Hopf algebras and Markov chains: two examples and a theory
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Abstract
The operation of squaring (coproduct followed by product) in a combinatorial Hopf algebra is shown to induce a Markov chain in natural bases. Chains constructed in this way include widely studied methods of card shuffling, a natural “rockbreaking” process, and Markov chains on simplicial complexes. Many of these chains can be explicitly diagonalized using the primitive elements of the algebra and the combinatorics of the free Lie algebra. For card shuffling, this gives an explicit description of the eigenvectors. For rockbreaking, an explicit description of the quasistationary distribution and sharp rates to absorption follow.
Keywords
Hopf Algebras Free Lie algebras Rock breaking models Shuffling1 Introduction
A Hopf algebra is an algebra \(\mathcal{H}\) with a coproduct \(\Delta:\mathcal{H}\to\mathcal{H}\otimes\mathcal{H}\) which fits together with the product \(m:\mathcal{H}\otimes\mathcal{H}\rightarrow\mathcal{H}\). Background on Hopf algebras is in Sect. 2.2. The map \(m\Delta:\mathcal{H}\to\mathcal{H}\) is called the Hopfsquare (often denoted Ψ ^{2} or x ^{[2]}). Our first discovery is that the coefficients of x ^{[2]} in natural bases can often be interpreted as a Markov chain. Specializing to familiar Hopf algebras can give interesting Markov chains: the free associative algebra gives the Gilbert–Shannon–Reeds model of riffle shuffling. Symmetric functions give a rockbreaking model of Kolmogoroff [54]. These two examples are developed first for motivation.
Example 1.1
(Free associative algebra and riffle shuffling)
Example 1.2
(Symmetric functions and rockbreaking)
A similar development works for any Hopf algebra which is either a polynomial algebra as an algebra (for instance, the algebra of symmetric functions, with generators e _{ n }), or is cocommutative and a free associative algebra as an algebra (e.g., the free associative algebra), provided each object of degree greater than one can be broken nontrivially. These results are described in Theorem 3.4.
Our second main discovery is that this class of Markov chains can be explicitly diagonalized using the Eulerian idempotent and some combinatorics of the free associative algebra. This combinatorics is reviewed in Sect. 2.3. It leads to a description of the left eigenvectors (Theorems 3.15 and 3.16) which is often interpretable and allows exact and asymptotic answers to natural probability questions. For a polynomial algebra, we are also able to describe the right eigenvectors completely (Theorem 3.19).
Example 1.3
(Shuffling)
Our results work for decks with repeated values allowing us to treat cases when, e.g., the suits do not matter and all picture cards are equivalent to tens. Here, fewer shuffles are required to achieve stationarity. For decks of essentially any composition we show that all eigenvalues 1/2^{ i }, 0≤i≤n−1, occur and determine multiplicities and eigenvectors.
Example 1.4
(Rockbreaking)
Section 2 reviews Markov chains (including uses for eigenvectors), Hopf algebras, and some combinatorics of the free associative algebra. Section 3 gives our basic theorems, generalizing the two examples to polynomial Hopf algebras and cocommutative, free associative Hopf algebras. Section 4 treats rockbreaking; Section 5 treats shuffling. Section 6 briefly describes other examples (e.g., graphs and simplicial complexes), counterexamples (e.g., the Steenrod algebra), and questions (e.g., quantum groups).
Two historical notes: The material in the present paper has roots in work of Patras [65, 66, 67], whose notation we are following, and Drinfeld [34]. Patras studied shuffling in a purely geometric fashion, making a ring out of polytopes in \(\mathbb{R}^{n}\). This study led to natural Hopf structures, Eulerian idempotents, and generalization of Solomon’s descent algebra in a Hopf context. His Eulerian idempotent maps decompose a graded commutative or cocommutative Hopf algebra into eigenspaces of the ath Hopfpowers; we improve upon this result, in the case of polynomial algebras or cocommutative, free associative algebras, by algorithmically producing a full eigenbasis. While there is no hint of probability in the work of Patras, it deserves to be much better known. More detailed references are given elsewhere in this paper.
We first became aware of Drinfeld’s ideas through their mention in Shnider–Sternberg [82]. Consider the Hopfsquare, acting on a Hopf algebra \(\mathcal{H}\). Suppose that \(x\in\mathcal{H}\) is primitive, Δ(x)=1⊗x+x⊗1. Then mΔ(x)=2x so x is an eigenvector of mΔ with eigenvalue 2. If x and y are primitive then mΔ(xy+yx)=4(xy+yx) and, similarly, if x _{1},…,x _{ k } are primitive then the sum of symmetrized products is an eigenvector of mΔ with eigenvector 2^{ k }. Drinfeld [34, Prop. 3.7] used these facts without comment in his proof that any formal deformation of the cocommutative universal enveloping algebra \(\mathcal{U}(\mathfrak{g})\) results already from deformation of the underlying Lie algebra \(\mathfrak{g}\). See [82, Sect. 3.8] and Sect. 3.4 below for an expanded argument and discussion. For us, a description of the primitive elements and their products gives the eigenvectors of our various Markov chains. This is developed in Sect. 3.
2 Background
This section gives notation and background for Markov chains (including uses for eigenvectors), Hopf algebras, the combinatorics of the free associative algebra and symmetric functions. All of these are large subjects and pointers to accessible literature are provided.
2.1 Markov chains
Throughout, we are in the unusual position of knowing β _{ i }, g _{ i } and possibly f _{ i } explicitly. This is rare enough that some indication of the use of eigenfunctions is indicated.
Use A
Use B
Use C
For f a right eigenfunction with eigenvalue β, let Y _{ i }=f(X _{ i })/β ^{ i }, 0≤i<∞. Then Y _{ i } is an \(\mathcal{F}_{i}\) martingale with \(\mathcal {F}_{i}=\sigma (X_{0},X_{1}, \ldots, X_{i} )\). One may try to use optional stopping, maximal and concentration inequalities and the martingale central limit theorem to study the behavior of the original X _{ i } chain.
Use D
One standard use of right eigenfunctions is to prove lower bounds for mixing times for Markov chains. The earliest use of this is the second moment method [26]. Here, one uses the second eigenfunction as a test function and expands its square in the eigenbasis to get concentration bounds. An important variation is Wilson’s method [95] which only uses the first eigenfunction but needs a careful understanding of the variation of this eigenfunction. A readable overview of both methods and many examples is in [76].
Use E
The left eigenfunctions come into computations since ∑_{ x } g _{ i }(x)f _{ j }(x)=δ _{ ij }. Thus in (2.1), a _{ i }=〈g _{ i }f/π〉. (Here f/π is just the density of f with respect to π.)
Use F
A second prevalent use of left eigenfunctions throughout this paper: the dual of a Hopf algebra is a Hopf algebra and left eigenfunctions of the dual chain correspond to right eigenfunctions of the original chain. This is similar to the situation for time reversal. If \(K^{*}(x,y)=\frac{\pi(y)}{\pi(x)}K(y,x)\) is the timereversed chain (note K ^{∗}(x,y) is a Markov chain with stationary distribution π), then g _{ i }/π is a right eigenfunction of K ^{∗}.
Use G
Use H
2.2 Hopf algebras
A Hopf algebra is an algebra \(\mathcal{H}\) over a field k (usually the real numbers in the present paper). It is associative with unit 1, but not necessarily commutative. Let us write m for the multiplication in \(\mathcal{H}\), so m(x⊗y)=xy. Then \(m^{[a]}:\mathcal{H}^{\otimes a}\to\mathcal{H}\) will denote afold products (so m=m ^{[2]}), formally m ^{[a]}=m(ι⊗m ^{[a−1]}) where ι denotes the identity map.
The product and coproduct have to be compatible so Δ is an algebra homomorphism, where multiplication on \(\mathcal{H}\otimes \mathcal{H}\) is componentwise; in Sweedler notation this says Δ(xy)=∑_{(x),(y)} x _{(1)} y _{(1)}⊗x _{(2)} y _{(2)}. All of the algebras considered here are graded and connected, i.e., \(\mathcal{H}=\bigoplus_{i=0}^{\infty}\mathcal{H}_{i}\) with \(\mathcal {H}_{0}=k\) and \(\mathcal{H}_{n}\) finitedimensional. The product and coproduct must respect the grading so \(\mathcal{H}_{i}\mathcal{H}_{j}\subseteq\mathcal{H}_{i+j}\), and \(x\in\mathcal{H}_{n}\) implies \(\Delta(x)\in\bigoplus_{j=0}^{n}\mathcal{H}_{j}\otimes\mathcal {H}_{nj}\). There are a few more axioms concerning a counit map and an antipode (automatic in the graded case); for the present paper, the most important is that the counit is zero on elements of positive degree, so, by the coalgebra axioms, \(\bar{\Delta}(x):=\Delta(x)  1 \otimes x  x \otimes1 \in\bigoplus_{j=1}^{n1}\mathcal{H}_{j}\otimes\mathcal{H}_{nj}\), for \(x\in\mathcal{H}_{n}\). The free associative algebra and the algebra of symmetric functions, discussed in Sect. 1, are examples of graded Hopf algebras.
The subject begins in topology when H. Hopf realized that the presence of the coproduct leads to nice classification theorems which allowed him to compute the cohomology of the classical groups in a unified manner. Topological aspects are still a basic topic [46] with many examples which may provide grist for the present mill. For example, the cohomology groups of the loops on a topological space form a Hopf algebra, and the homology of the loops on the suspension of a wedge of circles forms a Hopf algebra isomorphic to the free associative algebra of Example 1.1 [14].
Joni and Rota [49] realized that many combinatorial objects have a natural breaking structure which gives a coalgebra structure to the graded vector space on such objects. Often there is a compatible way of putting pieces together, extending this to a Hopf algebra structure. Often, either the assembling or the breaking process is symmetric, leading to commutative or cocommutative Hopf algebras, respectively. For example, the symmetric function algebra is commutative and cocommutative while the free associative algebra is just cocommutative.
The theory developed here is for graded commutative or cocommutative Hopf algebras with one extra condition: that there is a unique way to assemble any given collection of objects. This amounts to the requirement that the Hopf algebra is either a polynomial algebra as an algebra (and therefore commutative) or a free associative algebra as an algebra and cocommutative (and therefore noncommutative). (We write a free associate algebra to refer to the algebra structure only, as opposed to the free associative algebra which has a specified coalgebra structure—namely, the generating elements are primitive.)
Increasingly sophisticated developments of combinatorial Hopf algebras are described by [4, 77, 78, 79, 80] and [1]. This last is an expansive extension which unifies many common examples. Below are two examples that are prototypes for their Bosonic Fock functor and Full Fock functor constructions, respectively [1, Ch. 15]; they are also typical of constructions detailed in other sources.
Example 2.1
(The Hopf algebra of unlabeled graphs) [79, Sect. 12] [35, Sect. 3.2]
Example 2.2
(The noncommutative Hopf algebra of labeled graphs) [79, Sect. 13] [35, Sect. 3.3]
Aguiar–Bergeron–Sottile [4] define a combinatorial Hopf algebra as a Hopf algebra \(\mathcal{H}\) with a character \(\zeta :\mathcal{H}\to k\) which is both additive and multiplicative. They prove a universality theorem: any combinatorial Hopf algebra has a unique characterpreserving Hopf morphism into the algebra of quasisymmetric functions. They show that this unifies many ways of building generating functions. When applied to the Hopf algebra of graphs, their map gives the chromatic polynomial. In Sect. 3.7 we find that their map gives the probability of absorption for several of our Markov chains. See also the examples in Sect. 6.
A good introduction to Hopf algebras is in [82]. A useful standard reference is in [64] and our development does not use much outside of her Chap. 1. The broadranging text [62] is aimed towards quantum groups but contains many examples useful here. Quantum groups are neither commutative nor cocommutative and need special treatment; see Example 6.3.
A key ingredient in our work is the Hopfsquare map Ψ ^{2}=mΔ; Ψ ^{2}(x) is also written x ^{[2]}. In Sweedler notation, Ψ ^{2}(x)=∑_{(x)} x _{(1)} x _{(2)}; in our combinatorial setting, it is useful to think of “pulling apart” x according to Δ, then using the product to put the pieces together. On graded Hopf algebras, Ψ ^{2} preserves the grading and, appropriately normalized, gives a Markov chain on appropriate bases. See Sect. 3.2 for assumptions and details. The higher power maps Ψ ^{ a }=m ^{[a]}Δ^{[a]} will also be studied, since under our hypothesis, they present no extra difficulty. For example, Ψ ^{3}(x)=∑_{(x)} x _{(1)} x _{(2)} x _{(3)}. In the shuffling example, Ψ ^{ a } corresponds to the “ashuffles” of [10]. A theorem of [90] shows that, for commutative or cocommutative Hopf algebras, the power rule holds: (x ^{[a]})^{[b]}=x ^{[ab]}, or Ψ ^{ a } Ψ ^{ b }=Ψ ^{ ab }. See also the discussion in [56]. In shuffling language this becomes “an ashuffle followed by a bshuffle is an abshuffle” [10]. In general Hopf algebras this power law often fails [51]. Power maps are actively studied as part of a program to carry over to Hopf algebras some of the rich theory of groups. See [44, 59] and their references.
2.3 Structure theory of a free associative algebra
The eigenvectors of our Markov chains are described using combinatorics related to the free associative algebra, as described in the selfcontained [60, Chap. 5].

25413 has coefficient 1 in λ(13245) since the unique way to rearrange T _{13245} so the leaves spell 25413 is to exchange the branches at the root and the highest interior vertex;

21345 does not appear in λ(13245) since whenever the branches of T _{13245} switch, 2 must appear adjacent to either 4 or 5, which does not hold for 21345;

1221 has coefficient 0 in λ(1122) as, to make the leaves of T _{1122} spell 1221, we can either exchange branches at the root, or exchange branches at both of the other interior vertices. These two rearrangements have opposite signs, so the signed count of rearrangements is 0.
We can again express the coefficient of w′ in \(\operatorname {sym}(w)\) as the signed number of ways to rearrange T _{ w } so the leaves spell w′. Now there are two types of allowed moves: exchanging the left and right branches at a vertex, and permuting the trees of the hedgerow. The latter move does not come with a sign. Thus 14253 has coefficient −1 in \(\operatorname{sym}(35142)\), as the unique rearrangement of T _{35142} which spells 14253 requires transposing the trees and permuting the branches labeled 3 and 5.
It is clear from this pictorial description that every term appearing in \(\operatorname{sym}(w)\) is a permutation of the letters in w. Garsia and Reutenauer [37, Th. 5.2] shows that \(\{\operatorname{sym}(w)\}\) form a basis for a free associative algebra. This will turn out to be a left eigenbasis for inverse riffle shuffling, and similar theorems hold for other Hopf algebras.
2.4 Symmetric functions and beyond
A basic object of study is the vector space \(\varLambda_{k}^{n}\) of homogeneous symmetric polynomials in k variables of degree n. The direct sum \(\varLambda_{k}=\bigoplus_{n=0}^{\infty}\varLambda_{k}^{n}\) forms a graded algebra with familiar bases: the monomial (m _{ λ }), elementary (e _{ λ }), homogeneous (h _{ λ }), and power sums (p _{ λ }). For example, e _{2}(x _{1},…,x _{ k })=∑_{1≤i<j≤k } x _{ i } x _{ j } and for a partition λ=λ _{1}≥λ _{2}≥⋯≥λ _{ l }>0 with λ _{1}+⋯+λ _{ l }=n, \(e_{\lambda}=e_{\lambda_{1}}e_{\lambda_{2}}\cdots e_{\lambda_{l}}\). As λ ranges over partitions of n, {e _{ λ }} form a basis for \(\varLambda_{k}^{n}\), from which we construct the rockbreaking chain of Example 1.2. Splendid accounts of symmetric function theory appear in [61] and [86]. A variety of Hopf algebra techniques are woven into these topics, as emphasized by [38] and [96]. The comprehensive account of noncommutative symmetric functions [39] and its followups furthers the deep connection between combinatorics and Hopf algebras. However, this paper will only involve its dual, the algebra of quasisymmetric functions, as they encode informations about absorption rates of our chains, see Sect. 3.7. A basis of this algebra is given by the monomial quasisymmetric functions: for a composition α=(α _{1},…,α _{ k }), define \(M_{\alpha}=\sum_{i_{1} < i_{2} < \cdots<i_{k}} x_{i_{1}}^{\alpha_{1}} \cdots x_{i_{k}}^{\alpha_{k}}\). Further details are in [86, Sect. 7.19].
3 Theory
3.1 Introduction
This section states and proves our main theorems. This introduction sets out definitions. Section 3.2 develops the reweighting schemes needed to have the Hopfsquare maps give rise to Markov chains. Section 3.3 explains that these chains are often acyclic. Section 3.4 addresses a symmetrization lemma that we will use in Sections 3.5 and 3.6 to find descriptions of some left and right eigenvectors, respectively, for such chains. Section 3.7 determines the stationary distributions and gives expressions for the chance of absorption in terms of generalized chromatic polynomials. Applications of these theorems are in the last three sections of this paper.
 1.
\(\mathcal{H}=\mathbb{R} [c_{1},c_{2},\ldots ]\) as an algebra (i.e., \(\mathcal{H}\) is a polynomial algebra) and \(\mathcal {B}= \{ c_{1}^{n_{1}}c_{2}^{n_{2}}\cdots\mid n_{i}\in\mathbb{N} \}\), the basis of monomials. The c _{ i } may have any degree, and there is no constraint on the coalgebra structure. This will give rise to a Markov chain on combinatorial objects where assembling is symmetric and deterministic.
 2.
\(\mathcal{H}\) is cocommutative, \(\mathcal{H}=\mathbb{R}\langle c_{1},c_{2},\ldots \rangle \) as an algebra, (i.e., \(\mathcal{H}\) is a free associative algebra) and \(\mathcal{B}= \{ c_{i_{1}}c_{i_{2}}\cdots \mid i_{j}\in\mathbb{N} \}\), the basis of words. The c _{ i } may have any degree, and do not need to be primitive. This will give rise to a Markov chain on combinatorial objects where pulling apart is symmetric, assembling is nonsymmetric and deterministic.
Write \(\mathcal{H}_{n}\) for the subspace of degree n in \(\mathcal {H}\), and \(\mathcal{B}_{n}\) for the degree n basis elements. The generators c _{ i } can be identified as those basis elements which are not the nontrivial product of basis elements; in other words, generators cannot be obtained by assembling objects of lower degree. Thus, all basis elements of degree one are generators, but there are usually generators of higher degree; see Examples 3.1 and 3.2 below. One can view the conditions 1 and 2 above as requiring the basis elements to have unique factorization into generators, allowing the convenient view of \(b \in\mathcal{B}\) as a word b=c _{1} c _{2}⋯c _{ l }. Its length l(b) is then welldefined—it is the number of generators one needs to assemble together to produce b. Some properties of the length are developed in Sect. 3.3. For a noncommutative Hopf algebra, it is useful to choose a linear order on the set of generators refining the ordering by degree: i.e. if deg(c)<deg(c′), then c<c′. This allows the construction of the Lyndon factorization and standard bracketing of a basis element, as in Sect. 2.3. Example 3.17 demonstrates such calculations.
The ath Hopfpower map is Ψ ^{ a }:=m ^{[a]}Δ^{[a]}, the afold coproduct followed by the afold product. These power maps are the central object of study of [65, 66, 67]. Intuitively, Ψ ^{ a } corresponds to breaking an object into a pieces (some possibly empty) in all possible ways and then reassembling them. The Ψ ^{ a } preserve degree, thus mapping \(\mathcal{H}_{n}\) to \(\mathcal{H}_{n}\).
3.2 The Markov chain connection
The power maps can sometimes be interpreted as a natural Markov chain on the basis elements \(\mathcal{B}_{n}\) of \(\mathcal{H}_{n}\).
Example 3.1
(The Hopf algebra of unlabeled graphs, continuing from Example 2.1)
The resulting Markov chain on graphs with n vertices evolves as follows: from G, color the vertices of G red or blue, independently with probability 1/2. Erase any edge with opposite colored vertices. This gives one step of the chain; the process terminates when there are no edges. Observe that each connected component breaks independently; that Δ is an algebra homomorphism ensures that, for any Hopf algebra, the generators break independently. The analogous Hopf algebra of simplicial complexes is discussed in Sect. 6.
Example 3.2
(The noncommutative Hopf algebra of labeled graphs, continuing from Example 2.2)
When is such a probabilistic interpretation possible? To begin, the coefficients of mΔ(b) must be nonnegative real numbers for \(b\in\mathcal{B}\). This usually holds for combinatorial Hopf algebras, but the free associative algebra and the above algebras of graphs have an additional desirable property: for any \(b\in\mathcal {B}\), the coefficients of Ψ ^{2}(b) sum to 2^{deg(b)}, regardless of b. Thus the operator \(\frac{1}{2^{n}}\varPsi^{2}(b)=\sum_{b'}K(b,b')b'\) forms a Markov transition matrix on basis elements of degree n. Indeed, the coefficients of Ψ ^{ a }(b) sum to a ^{deg(b)} for all a, so \(\frac{1}{a^{n}}\varPsi^{a}(b)=\sum_{b'}K_{a}(b,b')b'\) defines a transition matrix K _{ a }. For other Hopf algebras, the sum of the coefficients in Ψ ^{2}(b) may depend on b, so simply scaling Ψ ^{2} does not always yield a transition matrix.
Zhou’s rephrasing [97, Lemma 4.4.1.1] of the Doob transform [58, Sect. 17.6.1] provides a solution: if K is a matrix with nonnegative entries and ϕ is a strictly positive right eigenfunction of K with eigenvalue 1, then \(\hat{K}(b,b'):= \phi(b)^{1}K(b,b')\phi(b')\) is a transition matrix. Here \(\hat{K}\) is the conjugate of K by the diagonal matrix whose entries are ϕ(b). Theorem 3.4 below gives conditions for such ϕ to exist, and explicitly constructs ϕ recursively; Corollary 3.5 then specifies a nonrecursive definition of ϕ when there is a sole basis element of degree 1. The following example explains why this construction is natural.
Example 3.3
(Symmetric functions and rockbreaking)
The following theorem shows that this algorithm works in many cases. Observe that, in the above example, it is the nonzero offdiagonal entries that change; the diagonal entries cannot be changed by rescaling the basis. Hence the algorithm would fail if some row had all offdiagonal entries equal to 0, and diagonal entry not equal to 1. This corresponds to the existence of \(b \in\mathcal{B}_{n}\) with \(\frac {1}{2^{n}}\varPsi^{2} (b)=\alpha b\) for some α≠1; the condition \(\bar{\Delta}(c):=\Delta(c)  1 \otimes c  c\otimes1 \neq0\) below precisely prevents this. Intuitively, we are requiring that each generator of degree greater than one can be broken nontrivially. For an example where this condition fails, see Example 6.5.
Theorem 3.4
(Basis rescaling)
Remarks
 1.
Observe that, if b=xy, then the definition of \(\hat{b}\) ensures \(\hat{b}=\hat{x}\hat{y}\). Equivalently, ϕ is a multiplicative function.
 2.
The definition of \(\hat{c}\) is not circular: since \(\mathcal {H}\) is graded with \(\mathcal{H}_{0}=\mathbb{R}\), the counit is zero on elements of positive degree so that \(\bar{\Delta}(c)\in\bigoplus_{j=1}^{\deg (c)1} \mathcal{H}_{j}\otimes \mathcal{H}_{\deg(c)j}\). Hence K _{2}(c,b) is nonzero only if b=c or l(b)>1, so the denominator in the expression for \(\hat{c}\) only involves ϕ(b) for b with l(b)>1. Such b can be factorized as b=xy with deg(x),deg(y)<deg(b), whence ϕ(b)=ϕ(x)ϕ(y), so \(\hat{c}\) only depends on ϕ(x) with deg(x)<deg(c).
Proof
The above showed each row of \(\hat{K}_{2}\) sums to 1, which means (1,1,…,1) is a right eigenvector of \(\hat{K}_{2}\) of eigenvalue 1. \(\hat{K}_{a}\) describes Ψ ^{ a } in the \(\hat{\mathcal{B}}\) basis, which is also a basis of monomials/words, in a rescaled set of generators \(\hat{c}\), so, by Theorems 3.19 and 3.20, the eigenspaces of \(\hat{K}_{a}\) do not depend on a. Hence (1,1,…,1) is a right eigenvector of \(\hat{K}_{a}\) of eigenvalue 1 for all a, thus each row of \(\hat{K}_{a}\) sums to 1 also.
Combinatorial Hopf algebras often have a single basis element of degree 1—for the algebra of symmetric functions, this is the unique partition of 1; for the Hopf algebra \(\mathcal{G}\) of graphs, this is the discrete graph with one vertex. After the latter example, denote this basis element by •. Then there is a simpler definition of the eigenfunction ϕ, and hence \(\hat{b}\) and \(\hat{K}\), in terms of \(\eta_{b}^{b_{1},\ldots,b_{r}}\), the coefficient of b _{1}⊗⋯⊗b _{ r } in Δ^{[r]}(b):
Corollary 3.5
Proof
Work on \(\mathcal{H}_{n}\) for a fixed degree n. Recall that ϕ is a right eigenvector of \(\hat{K}_{a}\) of eigenvalue 1, and hence, by the notation of Sect. 3.6, an eigenvector of Ψ ^{∗a } of eigenvalue a ^{ n }. By Theorems 3.19 and 3.20, this eigenspace is spanned by f _{ b } for b with length n. Then \(\mathcal{B}_{1} =\{\bullet\}\) forces b=•^{ n }, so \(f_{\bullet^{n}}(b')=\frac{1}{n!}\eta_{b'}^{\bullet,\ldots,\bullet}\) spans the a ^{ n }eigenspace of Ψ ^{∗a }. Consequently, ϕ is a multiple of \(f_{\bullet^{n}}\). To determine this multiplicative factor, observe that Theorem 3.4 defines ϕ(•) to be 1, so ϕ(•^{ n })=1, and \(f_{\bullet^{n}}(\bullet^{n})=1\) also, so \(\phi=f_{\bullet^{n}}\). □
3.3 Acyclicity
Observe that the rockbreaking chain (Examples 1.2 and 3.3) is acyclic—it can never return to a state it has left, because the only way to leave a state is to break the rocks into more pieces. More specifically, at each step the chain either stays at the same partition or moves to a partition which refines the current state; as refinement of partitions is a partial order, the chain cannot return to a state it has left. The same is true for the chain on unlabeled graphs (Example 3.1)—the number of connected components increases over time, and the chain never returns to a previous state. Such behavior can be explained by the way the length changes under the product and coproduct. (Recall that the length l(b) is the number of factors in the unique factorization of b into generators.) Define a relation on \(\mathcal{B}\) by b→b′ if b′ appears in Ψ ^{ a }(b) for some a. If Ψ ^{ a } induces a Markov chain on \(\mathcal{B}_{n}\), then this precisely says that b′ is accessible from b.
Lemma 3.6
 (i)
l(b _{1}⋯b _{ a })=l(b _{1})+⋯+l(b _{ a });
 (ii)
For any summand b _{(1)}⊗⋯⊗b _{(a)} in Δ^{[a]}(b), l(b _{(1)})+⋯+l(b _{(a)})≥l(b);
 (iii)
if b→b′, then l(b′)≥l(b).
Proof
(i) is clear from the definition of length.
Prove (ii) by induction on l(b). Note that the claim is vacuously true if b is a generator, as each l(b _{(i)})≥0, and not all l(b _{(i)}) may be zero. If b factorizes nontrivially as b=xy, then, as Δ^{[a]}(b)=Δ^{[a]}(x)Δ^{[a]}(y), it must be the case that b _{(i)}=x _{(i)} y _{(i)}, for some x _{(1)}⊗⋯⊗x _{(a)} in Δ^{[a]}(x), y _{(1)}⊗⋯⊗y _{(a)} in Δ^{[a]}(y). So l(b _{(1)})+⋯+l(b _{(a)})=l(x _{(1)})+⋯+l(x _{(a)})+l(y _{(1)})+⋯+l(y _{(a)}) by (i), and by inductive hypothesis, this is at least l(x)+l(y)=l(b).
(iii) follows trivially from (i) and (ii): if b→b′, then b′=b _{(1)}⋯b _{(a)} for a term b _{(1)}⊗⋯⊗b _{(a)} in Δ^{[a]}(b). So l(b′)=l(b _{(1)})+⋯+l(b _{(a)})≥l(b). □
If \(\mathcal{H}\) is a polynomial algebra, more is true. The following proposition explains why chains built from polynomial algebras (i.e., with deterministic and symmetric assembling) are always acyclic; in probability language, it says that, if the current state is built from l generators, then, with probability a ^{ l−n }, the chain stays at this state, otherwise, it moves to a state built from more generators. Hence, if the states are totally ordered to refine the partial ordering by length, then the transition matrices are uppertriangular with a ^{ l−n } on the main diagonal.
Proposition 3.7
(Acyclicity)
Proof
It is easier to first prove the expression for Ψ ^{ a }(b). Suppose b has factorization into generators b=c _{1} c _{2}⋯c _{ l(b)}. As \(\mathcal{H}\) is commutative, Ψ ^{ a } is an algebra homomorphism, so Ψ ^{ a }(b)=Ψ ^{ a }(c _{1})⋯Ψ ^{ a }(c _{ l(b)}). Recall from Sect. 2.2 that \(\bar{\Delta}(c)=\Delta(c)1 \otimes c  c\otimes 1 \in\bigoplus_{i=1}^{deg(c)1}\mathcal{H}_{i} \otimes\mathcal {H}_{deg(c)i}\), in other words, 1⊗c and c⊗1 are the only terms in Δ(c) which have a tensorfactor of degree 0. As Δ^{[3]}=(ι⊗Δ)Δ, the only terms in Δ^{[3]}(c) with two tensorfactors of degree 0 are 1⊗1⊗c, 1⊗c⊗1 and c⊗1⊗1. Inductively, we see that the only terms in Δ^{[a]}(c) with all but one tensorfactor having degree 0 are 1⊗⋯⊗1⊗c,1⊗⋯⊗1⊗c⊗1,…,c⊗1⊗⋯⊗1. So Ψ ^{ a }(c)=ac+∑_{ l(b′)>1} α _{ cb′} b′ for generators c. As Ψ ^{ a }(b)=Ψ ^{ a }(c _{1})⋯Ψ ^{ a }(c _{ l }), and length is multiplicative (Lemma 3.6(i)), the expression for Ψ ^{ a }(b) follows.
It is then clear that → is reflexive and antisymmetric. Transitivity follows from the power rule: if b→b′ and b′→b″, then b′ appears in Ψ ^{ a }(b) for some a and b″ appears in Ψ ^{ a′}(b′) for some a′. So b″ appears in Ψ ^{ a′} Ψ ^{ a }(b)=Ψ ^{ a′a }(b). □
Here is one more result in this spirit, necessary in Sect. 3.5 to show that the eigenvectors constructed there have good triangularity properties and hence form an eigenbasis:
Lemma 3.8
Let \(b,b_{i}, b_{i}'\) be monomials/words in a Hopf algebra which is either a polynomial algebra or a free associative algebra that is cocommutative. If b=b _{1}⋯b _{ k } and \(b_{i} \rightarrow b'_{i}\) for each i, then \(b \rightarrow b'_{\sigma(1)} \cdots b'_{\sigma(k)}\) for any σ∈S _{ k }.
Proof
For readability, take k=2 and write b=xy, x→x′, y→y′. By definition of the relation →, it must be that x′=x _{(1)}⋯x _{(a)} for some summand x _{(1)}⊗⋯⊗x _{(a)} of \(\bar{\Delta}^{[a]}(x)\). Likewise y′=y _{(1)}⋯y _{(a′)} for some a′. Suppose a>a′. Coassociativity implies that Δ^{[a]}(y)=(ι⊗⋯⊗ι⊗Δ^{[a−a′]})Δ^{[a′]}(y), and y _{(a′)}⊗1⊗⋯⊗1 is certainly a summand of Δ^{[a−a′]}(y _{(a′)}), so y _{(1)}⊗⋯⊗y _{(a′)}⊗1⊗⋯⊗1 occurs in Δ^{[a]}(y). So, taking y _{(a′+1)}=⋯=y _{(a)}=1, we can assume a=a′. Then Δ^{[a]}(b)=Δ^{[a]}(x)Δ^{[a]}(y) contains the term x _{(1)} y _{(1)}⊗⋯⊗x _{(a)} y _{(a)}. Hence Ψ ^{ a }(b) contains the term x _{(1)} y _{(1)}⋯x _{(a)} y _{(a)}, and this product is x′y′ if \(\mathcal{H}\) is a polynomial algebra.
If \(\mathcal{H}\) is a cocommutative, free associative algebra, the factors in x _{(1)} y _{(1)}⊗⋯⊗x _{(a)} y _{(a)} must be rearranged to conclude that b→x′y′ and b→y′x′. Coassociativity implies Δ^{[2a]}=(Δ⊗⋯⊗Δ)Δ^{[a]}, and Δ(x _{(i)} y _{(i)})=Δ(x _{(i)})Δ(y _{(i)}) contains (x _{(i)}⊗1)(1⊗y _{(i)})=x _{(i)}⊗y _{(i)}, so Δ^{[2a]}(b) contains the term x _{(1)}⊗y _{(1)}⊗x _{(2)}⊗y _{(2)}⊗⋯⊗x _{(a)}⊗y _{(a)}. As \(\mathcal{H}\) is cocommutative, any permutation of the tensorfactors, in particular, x _{(1)}⊗x _{(2)}⊗⋯⊗x _{(a)}⊗y _{(1)}⊗⋯⊗y _{(a)} and y _{(1)}⊗y _{(2)}⊗⋯⊗y _{(a)}⊗x _{(1)}⊗⋯⊗x _{(a)}, must also be summands of Δ^{[2a]}(b), and multiplying these tensorfactors together shows that both x′y′ and y′x′ appear in Ψ ^{[2a]}(b). □
Example 3.9
(Symmetric functions and rockbreaking)
Recall from Example 3.3 the algebra of symmetric functions with basis {e _{ λ }}, which induces the rockbreaking process. Here, e _{ λ }→e _{ λ′} if and only if λ′ refines λ. Lemma 3.8 for the case k=2 is the statement that, if λ is the union of two partitions μ and ν, and μ′ refines μ, ν′ refines ν, then μ′∐ν′ refines μ∐ν=λ.
3.4 The symmetrization lemma
 (i)
Make an eigenvector of smallest eigenvalue for each generator c;
 (ii)
For each basis element b with factorization c _{1} c _{2}⋯c _{ l }, build an eigenvector of larger eigenvalue out of the eigenvectors corresponding to the factors c _{ i }, produced in the previous step.
Concentrate on the left eigenvectors for the moment. Recall that the transition matrix K _{ a } is defined by a ^{−n } Ψ ^{ a }(b)=∑_{ b′} K _{ a }(b,b′)b′, so the left eigenvectors for our Markov chain are the usual eigenvectors of Ψ ^{ a } on \(\mathcal{H}\). Step (ii) is simple if \(\mathcal{H}\) is a polynomial algebra, because then \(\mathcal{H}\) is commutative so Ψ ^{ a } is an algebra homomorphism. Consequently, the product of two eigenvectors is an eigenvector with the product eigenvalue. This fails for cocommutative, free associative algebras \(\mathcal{H}\), but can be fixed by taking symmetrized products:
Theorem 3.10
(Symmetrization lemma)
Let x _{1},x _{2},…,x _{ k } be primitive elements of any Hopf algebra \(\mathcal{H}\), then \(\sum_{\sigma\in S_{k}} x_{\sigma(1)} x_{\sigma (2)} \cdots x_{\sigma(k)}\) is an eigenvector of Ψ ^{ a } with eigenvalue a ^{ k }.
Proof
In Sects. 3.5 and 3.6, the fact that the eigenvectors constructed give a basis will follow from triangularity arguments based on Sect. 3.3. These rely heavily on the explicit structure of a polynomial algebra or a free associative algebra. Hence it is natural to look for alternatives that will generalize this eigenbasis construction plan to Hopf algebras with more complicated structures. For example, one may ask whether some good choice of x _{ i } exists with which the symmetrization lemma will automatically generate a full eigenbasis. When \(\mathcal{H}\) is cocommutative, an elegant answer stems from the following two wellknown structure theorems:
Theorem 3.11
(Cartier–Milnor–Moore) [19, 63]
If \(\mathcal{H}\) is graded, cocommutative and connected, then \(\mathcal{H}\) is Hopf isomorphic to \(\mathcal {U}(\mathfrak{g})\), the universal enveloping algebra of a Lie algebra \(\mathfrak{g}\), where \(\mathfrak{g}\) is the Lie algebra of primitive elements of \(\mathcal{H}\).
Theorem 3.12
(Poincaré–Birkoff–Witt) [48, 60]
If {x _{1},x _{2},…} is a basis for a Lie algebra \(\mathfrak{g}\), then the symmetrized products \(\sum_{\sigma \in S_{k}} x_{i_{\sigma(1)}} x_{i_{\sigma(2)}} \cdots x_{i_{\sigma(k)}}\), for 1≤i _{1}≤i _{2}≤⋯≤i _{ k }, form a basis for \(\mathcal {U}(\mathfrak{g})\).
Putting these together reduces the diagonalization of Ψ ^{ a } on a cocommutative Hopf algebra to determining a basis of primitive elements:
Theorem 3.13
(Strong symmetrization lemma)
Much work [2, 3, 35] has been done on computing a basis for the subspace of the primitives of particular Hopf algebras, their formulas are in general more efficient than our universal method here, and using these will be the subject of future work. Alternatively, the theory of good Lyndon words [55] gives a Grobner basis argument to further reduce the problem to finding elements which generate the Lie algebra of primitives, and understanding the relations between them. This is the motivation behind our construction of the eigenvectors in Theorem 3.16, although the proof is independent of this theorem, more analogous to that of Theorem 3.15, the case of a polynomial algebra.
3.5 Left eigenfunctions
This map e is the first of a series of Eulerian idempotents e _{ i } defined by Patras [66]; he proves that, in a commutative or cocommutative Hopf algebra of characteristic zero where \(\bar{\Delta }\) is locally nilpotent (i.e. for each x, there is some a with \(\bar{\Delta}^{[a]} x=0\)), the Hopfpowers are diagonalizable, and these e _{ i } are orthogonal projections onto the eigenspaces. In particular, this weight decomposition holds for graded commutative or cocommutative Hopf algebras. We will not need the full series of Eulerian idempotents, although Example 3.18 makes the connection between them and our eigenbasis.
To deduce that the eigenvectors we construct are triangular with respect to \(\mathcal{B}\), one needs the following crucial observation (recall from Sect. 3.3 that b→b′ if b′ occurs in Ψ ^{ a }(b) for some a):
Proposition 3.14
Proof
The summand \(\frac{(1)^{a1}}{a}m^{[a]}\bar{\Delta}^{[a]}(c)\) involves terms of length at least a, from which the second expression of e(c) is immediate. Each term b′ of e(c) appears in Ψ ^{ a }(c) for some a, hence c→b′. Combine this with the knowledge from the second expression that c occurs with coefficient 1 to deduce the first expression. □
The two theorems below detail the construction of an eigenbasis for Ψ ^{ a } in a polynomial algebra and in a cocommutative free associative algebra, respectively. These are left eigenvectors for the corresponding transition matrices. A worked example will follow immediately; it may help to read these together.
Theorem 3.15
Theorem 3.16

for c a generator, set g _{ c }:=e(c);

for b a Lyndon word, inductively define \(g_{b}: = [g_{b_{1}}, g_{b_{2}} ]\) where b=b _{1} b _{2} is the standard factorization of b;

for b with Lyndon factorization b=b _{1}⋯b _{ k }, set \(g_{b}:=\sum_{\sigma\in S_{k}} g_{b_{\sigma(1)}} g_{b_{\sigma(2)}}\cdots g_{b_{\sigma(k)}}\).
Remarks
 1.
If Ψ ^{ a } defines a Markov chain, then the triangularity of g _{ b } (in both theorems) has the following interpretation: the left eigenfunction g _{ b } takes nonzero values only on states that are reachable from b.
 2.
The expression of the multiplicity of the eigenvalues (in both theorems) holds for Hopf algebras that are multigraded, if we replace all xs, ns and is by tuples, and read the formula as multiindex notation. For example, for a bigraded polynomial algebra \(\mathcal{H}\), the multiplicity of the a ^{ l }eigenspace in \(\mathcal{H}_{m,n}\) is the coefficient of \(x_{1}^{m} x_{2}^{n} y^{l}\) in \(\prod_{i,j} (1yx_{1}^{i} x_{2}^{j} )^{d_{i,j}}\), where d _{ i,j } is the number of generators of bidegree (i,j). This idea will be useful in Sect. 5.
 3.
Theorem 3.16 essentially states that any cocommutative free associative algebra is in fact isomorphic to the free associative algebra, generated by e(c). But there is no analogous interpretation for Theorem 3.15; being a polynomial algebra is not a strong enough condition to force all Hopf algebras with this condition to be isomorphic. A polynomial algebra \(\mathcal{H}\) is isomorphic to the usual polynomial Hopf algebra (i.e. with primitive generators) only if \(\mathcal{H}\) is cocommutative; then e(c) gives a set of primitive generators.
Example 3.17
Proof of Theorem 3.15 (polynomial algebra)
By Patras [66], the Eulerian idempotent map is a projection onto the aeigenspace of Ψ ^{ a }, so, for each generator c, e(c) is an eigenvector of eigenvalue a. As \(\mathcal{H}\) is commutative, Ψ ^{ a } is an algebra homomorphism, so the product of two eigenvectors is another eigenvector with the product eigenvalue. Hence g _{ b }:=e(c _{1})e(c _{2})⋯e(c _{ l }) is an eigenvector of eigenvalue a ^{ l }.
The multiplicity of the eigenvalue a ^{ l } is the number of basis elements b with length l. The last assertion of the theorem is then immediate from [94, Th. 3.14.1]. □
Example 3.18
It remains to show that the coefficient of b in e _{ l(b)}(b) is 1. Let b=c _{1}⋯c _{ l } be the factorization of b into generators. With notation from the previous paragraph, taking b′=b results in \(b \rightarrow b_{(1)} \cdots b_{(l)} \rightarrow b_{(1)}' \cdots b_{(l)}' =b\), so b=b _{(1)}⋯b _{(l)}. This forces the b _{(i)}=c _{ σ(i)} for some σ∈S _{ l }. As b _{(i)} occurs with coefficient 1 in e(b _{(i)}), the coefficient of b _{(1)}⊗⋯⊗b _{(l)} in (e⊗⋯⊗e)Δ^{[l]}(b) is the coefficient of c _{ σ(1)}⊗⋯⊗c _{ σ(l)} in Δ^{[l]}(b)=Δ^{[l]}(c _{1})⋯Δ^{[l]}(c _{ l }), which is 1 for each σ∈S _{ l }. Each occurrence of c _{ σ(1)}⊗⋯⊗c _{ σ(l)} in (e⊗⋯⊗e)Δ^{[l]}(b) gives rise to a b term in m ^{[l]}(e⊗e⊗⋯⊗e)Δ^{[l]}(b) with the same coefficient, for each σ∈S _{ l }, hence b has coefficient l! in m ^{[l]}(e⊗e⊗⋯⊗e)Δ^{[l]}(b)=l!e _{ l }(b).
The same argument also shows that, if i<l(b), then e _{ i }(b)=0, as there is no term of length i in e _{ i }(b). In particular, e(b)=0 if b is not a generator.
Proof of Theorem 3.16 (cocommutative and free associative algebra)
Schmitt [79, Thm. 9.4] shows that the Eulerian idempotent map e projects a graded cocommutative algebra onto its subspace of primitive elements, so g _{ c }:=e(c) is primitive. A straightforward calculation shows that, if \(x,y\in\mathcal{H}\) are primitive, then so is [x,y]. Iterating this implies that, if b is a Lyndon word, then g _{ b } (which is the standard bracketing of e(c)s) is primitive. Now apply the symmetrization lemma (Lemma 3.10) to deduce that, if \(b\in\mathcal{B}\) has k Lyndon factors, g _{ b } is an eigenvector of eigenvalue a ^{ k }.
The multiplicity of the eigenvalue a ^{ k } is the number of basis elements with k Lyndon factors. The last assertion of the theorem is then immediate from [94, Th. 3.14.1]. □
3.6 Right eigenvectors
The two theorems below give the eigenvectors of Ψ ^{∗a }; exemplar computations are in Sect. 4.2. Theorem 3.19 gives a complete description of these for \(\mathcal{H}\) a polynomial algebra, and Theorem 3.20 yields a partial description for \(\mathcal{H}\) a cocommutative free associative algebra. Recall that \(\eta_{b}^{b_{1},\ldots,b_{a}}\) is the coefficient of b _{1}⊗⋯⊗b _{ a } in Δ^{[a]}(b).
Theorem 3.19
Remarks
 1.
If \(\mathcal{H}\) is also cocommutative, then it is unnecessary to symmetrize—just define \(f_{b}=\frac {1}{A(b)}c_{1}^{*}c_{2}^{*}\cdots c_{l}^{*}\).
 2.
If Ψ ^{ a } defines a Markov chain on \(\mathcal{B}_{n}\), then the theorem says f _{ b }(b′) may be interpreted as the number of ways to break b′ into l pieces so that the result is some permutation of the l generators that are factors of b. In particular, f _{ b } takes only nonnegative values, and f _{ b } is nonzero only on states which can reach b. Thus f _{ b } may be used to estimate the probability of being in states that can reach b, see Corollary 4.10 for an example.
Theorem 3.20
Proof of Theorem 3.19 (polynomial algebra)
Suppose b ^{∗}⊗b′^{∗} is a term in m ^{∗}(c ^{∗}), where c is a generator. This means m ^{∗}(c ^{∗})(b⊗b′) is nonzero. Since comultiplication in \(\mathcal{H}^{*}\) is dual to multiplication in \(\mathcal{H}\), m ^{∗}(c ^{∗})(b⊗b′)=c ^{∗}(bb′), which is only nonzero if bb′ is a (real number) multiple of c. Since c is a generator, this can only happen if one of b,b′ is c. Hence c ^{∗} is primitive. Apply the symmetrization lemma (Lemma 3.10) to the primitives \(c_{1}^{*}, \ldots, c_{l}^{*}\) to deduce that f _{ b } as defined above is an eigenvector of eigenvalue a ^{ l }.
Since multiplication in \(\mathcal{H}^{*}\) is dual to the coproduct in \(\mathcal{H}\), \(c_{\sigma(1)}^{*} c_{\sigma(2)}^{*}\cdots c_{\sigma(l)}^{*}(b')=c_{\sigma (1)}^{*} \otimes c_{\sigma(2)}^{*} \otimes\cdots\otimes c_{\sigma(l)}^{*} (\Delta^{[l]}(b') )\), from which the first expression for f _{ b }(b′) is immediate. To deduce the second expression, note that the size of the stabilizer of (c _{ σ(1)},c _{ σ(2)},…,c _{ σ(l)}) under S _{ l } is precisely A(b).
It is apparent from the formula that b′^{∗} appears in f _{ b } only if c _{ σ(1)}⋯c _{ σ(l)}=b appears in Ψ ^{ l }(b′), hence b′→b is necessary. To calculate the leading coefficient f _{ b }(b), note that this is the sum over S _{ l } of the coefficients of c _{ σ(1)}⊗⋯⊗c _{ σ(l)} in Δ^{[l]}(b)=Δ^{[l]}(c _{1})⋯Δ^{[l]}(c _{ l }). Each term in Δ^{[l]}(c _{ i }) contributes at least one generator to at least one tensorfactor, and each tensorfactor of c _{ σ(1)}⊗⋯⊗c _{ σ(l)} is a single generator, so each occurrence of c _{ σ(1)}⊗⋯⊗c _{ σ(l)} is a product of terms from Δ^{[l]}(c _{ i }) where one tensorfactor is c _{ i } and all other tensorfactors are 1. Such products are all l! permutations of the c _{ i } in the tensorfactors, so, for each fixed σ, the coefficient of c _{ σ(1)}⊗⋯⊗c _{ σ(l)} in Δ^{[l]}(b) is A(b). This proves the first equality in the triangularity statement. Triangularity of f _{ b } with respect to length follows, as ordering by length refines the relation → (Proposition 3.7).
Proof of Theorem 3.20 (cocommutative and free associative algebra)
\(\mathcal{H}^{*}\) is commutative, so the power map is an algebra homomorphism. Then, since f _{ b } is defined as the product of k eigenvectors each of eigenvalue a, f _{ b } is an eigenvector of eigenvalue a ^{ k }.
The final statement is proved in the same way as in Theorem 3.19, for a polynomial algebra, since, when b=c _{1} c _{2}⋯c _{ l } with c _{1}≥c _{2}≥⋯≥c _{ l } in the ordering of generators, \(f_{b}=\frac{1}{A'(b)}c_{1}^{*}c_{2}^{*}\cdots c_{l}^{*}\). □
3.7 Stationary distributions, generalized chromatic polynomials, and absorption times
This section returns to probabilistic considerations, showing how the left eigenvectors of Sect. 3.5 determine the stationary distribution of the associated Markov chain. In the absorbing case, “generalized chromatic symmetric functions”, based on the universality theorem in [4], determine rates of absorption. Again, these general theorems are illustrated in the three sections that follow.
3.7.1 Stationary distributions
The first proposition identifies all the absorbing states when \(\mathcal{H}\) is a polynomial algebra:
Proposition 3.21
Suppose \(\mathcal{H}\) is a polynomial algebra where K _{ a }, defined by a ^{−n } Ψ ^{ a }(b)=∑_{ b′} K _{ a }(b,b′)b′, is a transition matrix. Then the absorbing states are the basis elements \(b\in\mathcal{B}_{n}\) which are products of n (possibly repeated) degree one elements, and these give a basis of the 1eigenspace of K _{ a }.
Example 3.22
In the commutative Hopf algebra of graphs in Examples 2.1 and 3.1, there is a unique basis element of degree 1—the graph with a single vertex. Hence the product of n such, which is the empty graph, is the unique absorbing state. Similarly for the rockbreaking example (symmetric functions) on partitions of n, the only basis element of degree 1 is e _{1} and the stationary distribution is absorbing at 1^{ n } (or \(e_{1^{n}}\)).
The parallel result for a cocommutative free associative algebra picks out the stationary distributions:
Proposition 3.23
Suppose \(\mathcal{H}\) is a cocommutative and free associative algebra where K _{ a }, defined by a ^{−n } Ψ ^{ a }(b)=∑_{ b′} K _{ a }(b,b′)b′, is a transition matrix. Then, for each unordered ntuple {c _{1},c _{2},…,c _{ n }} of degree 1 elements (some c _{ i } s may be identical), the uniform distribution on {c _{ σ(1)} c _{ σ(2)}⋯c _{ σ(n)}∣σ∈S _{ n }} is a stationary distribution for the associated chain. In particular, all absorbing states have the form •^{ n }, where \(\bullet\in\mathcal{B}_{1}\).
Example 3.24
In the free associative algebra \(\mathbb{R}\langle x_{1},x_{2},\ldots,x_{n}\rangle\), each x _{ i } is a degree 1 element. So the uniform distribution on x _{ σ(1)}⋯x _{ σ(n)} (σ∈S _{ n }) is a stationary distribution, as evident from considering inverse shuffles.
Proof of Proposition 3.21
From Theorem 3.15, a basis for the 1eigenspace is {g _{ b }∣l(b)=n}. This forces each factor of b to have degree 1, so b=c _{1} c _{2}⋯c _{ n } and g _{ b }=e(c _{1})⋯e(c _{ n }). Now \(e(c)=\sum_{a\geq1}\frac{(1)^{a1}}{a}m^{[a]}\bar{\Delta }^{[a]}(c)\), and, when deg(c)=1, \(m^{[a]}\bar{\Delta}^{[a]}(c)=0\) for all a≥2. So e(c)=c, and hence g _{ b }=c _{1} c _{2}⋯c _{ n }=b, which is a point mass on b, so b=c _{1} c _{2}⋯c _{ n } is an absorbing state. □
Proof of Proposition 3.23
From Theorem 3.16, a basis for the 1eigenspace is {\(g _{b} \mid b\in\mathcal{B}_{n}, b\) has n Lyndon factors}. This forces each Lyndon factor of b to have degree 1, so each of these must in fact be a single letter of degree 1. Thus b=c _{1} c _{2}⋯c _{ n } and \(g_{b}=\sum_{\sigma\in S_{n}} g_{c_{\sigma(1)}}\cdots g_{c_{\sigma(n)}}=\sum_{\sigma\in S_{n}} c_{\sigma(1)}\cdots c_{\sigma(n)}\), as g _{ c }=c for a generator c. An absorbing state is a stationary distribution which is a point mass. This requires c _{ σ(1)}⋯c _{ σ(n)} to be independent of σ. As \(\mathcal{H}\) is a free associative algebra, this only holds when c _{1}=⋯=c _{ n }=:•, in which case g _{ b }=n!•^{ n }, so •^{ n } is an absorbing state. □
3.7.2 Absorption and chromatic polynomials
Consider the case where there is a single basis element of degree 1; call this element • as in Sect. 3.2. Then, by Propositions 3.21 and 3.23, the K _{ a } chain has a unique absorbing basis vector \(\bullet^{n}\in\mathcal {H}_{n}\). The chance of absorption after k steps can be rephrased in terms of an analog of the chromatic polynomial. Note first that the property K _{ a }∗K _{ a′}=K _{ aa′} implies it is enough to calculate K _{ a }(b,•^{ n }) for general a and starting state \(b\in\mathcal {H}_{n}\). To do this, make \(\mathcal{H}\) into a combinatorial Hopf algebra in the sense of [4] by defining a character ζ that takes value 1 on • and value 0 on all other generators, and extend multiplicatively and linearly. In other words, ζ is an indicator function of absorption, taking value 1 on all absorbing states and 0 on all other states. By [4, Th. 4.1] there is a unique characterpreserving Hopf algebra map from \(\mathcal{H}\) to the algebra of quasisymmetric functions. Define χ _{ b } to be the quasisymmetric function that is the image of the basis element b under this map. (If \(\mathcal{H}\) is cocommutative, χ _{ b } will be a symmetric function.) Call this the generalized chromatic quasisymmetric function of b since it is the Stanley chromatic symmetric function for the Hopf algebra of graphs [84]. We do not know how difficult it is to determine or evaluate χ _{ b }.
Proposition 3.25
Proof
Using a different character ζ, this same argument gives the probability of reaching certain sets of states in one step of the K _{ a }(b,−) chain. This does not require \(\mathcal{H}\) to have a single basis element of degree 1.
Proposition 3.26
Example 3.27
(Rockbreaking)
Recall the rockbreaking chain of Example 3.3. Let \(\mathcal {C}= \{\hat{e}_{1}, \hat{e}_{2} \}\). Then \(\chi_{\hat {e}_{n}}^{\mathcal{C}} (2^{k},2^{k},\ldots,2^{k},0,0,\ldots )\) measures the probability that a rock of size n becomes rocks of size 1 or 2, after k binomial breaks.
4 Symmetric functions and breaking rocks
This section studies the Markov chain induced by the Hopf algebra of symmetric functions. Section 4.1 presents it as a rockbreaking process with background and references from the applied probability literature. Section 4.2 gives formulas for the right eigenfunctions by specializing from Theorem 3.19 and uses these to bound absorption time and related probabilistic observables. Section 4.3 gives formulas for the left eigenfunctions by specializing from Theorem 3.15 and uses these to derive quasistationary distributions.
4.1 Rockbreaking
The mathematical study of rockbreaking was developed by Kolmogoroff [54] who proved a log normal limit for the distribution of pieces of size at most x. A literature review and classical applied probability treatment in the language of branching processes is in [7]. They allow more general distributions for the size of pieces in each break. A modern manifestation with links to many areas of probability is the study of fragmentation processes. Extensive mathematical development and good pointers to a large physics and engineering literature are in [11, 12]. Most of the probabilistic development is in continuous time and has pieces breaking one at a time. We have not seen previous study of the natural model of simultaneous breaking developed here.
The rockbreaking Markov chain has the following alternative “balls in boxes” description:
Proposition 4.1
Remarks
Because of Proposition 4.1, many natural functions of the chain can be understood, as functions of k and n, from known properties of multinomial allocation. This includes the distribution of the size of the largest and smallest piece, the joint distribution of the number of pieces of size i, and total number of pieces. See [8, 53].
Observe from the matrices for n=2,3,4 that, if the partitions are written in reverselexicographic order, the transition matrices are lowertriangular. This is because lexicographic order refines the ordering of partitions by refinement. Furthermore, the diagonal entries are the eigenvalues 1/2^{ i }, as predicted in Proposition 3.7. Specializing the counting formula in Theorem 3.15 to this case proves the following proposition.
Proposition 4.2
The rockbreaking chain P _{ n }(λ,μ) on partitions of size n has eigenvalues 1/2^{ i }, 0≤i≤n−1, with 1/2^{ i } having multiplicity p(n,i), the number of partitions of n into i parts.
Note
As noted by the reviewer, the same rockbreaking chain can be derived using the basis {h _{ λ }} of complete homogeneous symmetric functions instead of {e _{ λ }}, since the two bases have the same product and coproduct structures. The fact that the dual basis to {h _{ λ }} is the more recognizable monomial symmetric functions {m _{ λ }}, compared to \(\{e_{\lambda}^{*}\}\), is unimportant here as the calculations of eigenfunctions do not explicitly require the dual basis.
4.2 Right eigenfunctions
Because the operator P _{ n }(λ,μ) is not selfadjoint in any reasonable sense, the left and right eigenfunctions must be developed distinctly. The following description, a specialization of Theorem 3.19, may be supplemented by the examples and corollaries that follow. A proof is at the end of this section.
Proposition 4.3
Here is an illustration of how to compute with this formula:
Example 4.4
Example 4.5
For some λ,μ pairs, the formula for f _{ μ }(λ) simplifies.
Example 4.6
Example 4.7
\(f_{1^{n}} \equiv1\) has eigenvalue 1.
Example 4.8
As (n) is not a refinement of any other partition, f _{(n)} is nonzero only at (n). Hence f _{(n)}(λ)=δ _{(n)}(λ).
Example 4.9
Example 4.9 can be applied to give bounds on the chance of absorption. The following corollary shows that absorption is likely after k=2log_{2} n+c steps.
Corollary 4.10
Proof
Remarks
 1.From Proposition 4.1 and the classical birthday problem, It follows that, for k=2log_{2} n+c (or 2^{ k }=2^{ c } n ^{2}) the inequality in the corollary is essentially an equality.
 2.Essentially the same calculations go through for any starting state λ, since by Use A$$E_\lambda \bigl\{f_\mu(X_k) \bigr\}= \frac{f_\mu(\lambda)}{2^k}. $$
 3.Other eigenfunctions can be similarly used. For example, when μ=1^{ n−r } r, \(f_{\mu}(\lambda)=\sum_{j}\binom{\lambda_{j}}{r}>0\) if and only if max_{ j } λ _{ j }≥r, and f _{ μ }(λ)≥1 otherwise. It follows as above that$$P_{(n)}\Bigl\{\max_i(X_k)_i \geq r\Bigr\}=P_{(n)}\bigl\{f_\mu(X_k)\geq1\bigr \}\leq E_{(n)}\bigl\{f_\mu(X_k)\bigr\}= \frac{\binom{n}{r}}{2^{(r1)k}}. $$
 4.The right eigenfunctions with μ=1^{ n−r } r can be derived by a direct probabilistic argument. Drop n balls into 2^{ k } boxes. Let N _{ i } be the number of balls in box i. ThenThe last equality follows because N _{1} is binomial (n,1/2^{ k }) and, if X is binomial (n,p), E{X(X−1)⋯(X−r+1)}=n(n−1)⋯(n−r+1)p ^{ r }. The other eigenvectors can be derived using more complicated multinomial moments.$$E_{(n)} \Biggl\{\sum_{i=1}^{2^k} \binom{N_i}{r} \Biggr\} =2^kE_{(n)} \biggl\{ \binom{N_1}{r} \biggr\}=\frac{\binom{n}{r}}{2^{(r1)k}}. $$
Proof of Proposition 4.3
4.3 Left eigenfunctions and quasistationary distributions
This section gives two descriptions of the left eigenfunctions: one in parallel with Proposition 4.3 and the other using symmetric function theory. Again, examples follow the statement with proofs at the end.
Proposition 4.11
As previously, here is a calculational example.
Example 4.12
Example 4.13
Observe that these matrices are the inverses of those in Example 4.5, as claimed in Theorem 3.19.
The next three examples give some partitions λ for which the expression for g _{ λ }(μ) condenses greatly:
Example 4.14
Example 4.15
Example 4.16
Take λ=1^{ n }. As no other partition refines λ, \(g_{1^{n}}(\mu)=\delta_{1^{n}}(\mu)\), and this is the stationary distribution.
The left eigenfunctions can be used to determine the quasistationary distributions π ^{1}, π ^{2} described in Sect. 2.1, Use G.
Corollary 4.17
Proof
From (2.4), π ^{1} is proportional to \(g_{1^{n2}2}\) on the nonabsorbing states. The Perron–Frobenius theorem ensures that π ^{1} is nonnegative. From Examples 4.15 and 4.16, \(\pi^{1}(\mu)=\delta_{1^{n2}2}(\mu)\) for μ≠1^{ n }. Similarly, π ^{2} is proportional to the pointwise product \(g_{1^{n2}2}(\mu)f_{1^{n2}2}(\mu)=\delta_{1^{n2}2}(\mu)\) for μ≠1^{ n }. □
For example, for λ=1^{ n }, p _{1}=e _{1}, and \(p_{1^{n}}=e_{1^{n}}=\hat{e}_{1^{n}}\) corresponding to the left eigenvector (1,0,…,0) with eigenvalue 1. For λ=21^{ n−2}, \(p_{2}=e_{1}^{2}2e_{2}=\hat{e}_{1}^{2}\hat{e}_{2}\), so \(p_{1^{n2}2}=\hat{e}_{1^{n}}\hat{e}_{1^{n2}2}\) corresponding to the eigenvector (1,−1,0,…,0) with eigenvalue 1/2. For λ=1^{ n−3}3, Open image in new window . Multiplying by 2 gives the left eigenvector (2,−3,1,0,…,0) with eigenvalue 1/4.
Proof of Proposition 4.11
Note
This calculation is greatly simplified for the algebra of symmetric functions, compared to other polynomial algebras. The reason is that, for a generator c, it is in general false that all terms of \(m^{[a]}\bar{\Delta}^{[a]}(c)\) have length a, or equivalently that all tensorfactors of a term of \(\bar{\Delta}^{[a]}(c)\) are generators. See the fourth summand of the coproduct calculation in Example 2.2 for an example. Then terms of length say, three, in e(c) may show up in both \(m^{[2]}\bar{\Delta}^{[2]}(c)\) and \(m^{[3]}\bar{\Delta }^{[3]}(c)\), so determining the coefficient of this length three term in e(c) is much harder, due to these potential cancelations in e(c). Hence much effort [2, 3, 35] has gone into developing cancelationfree expressions for primitives, as alternatives to e(c).
5 The free associative algebra and riffle shuffling
This section works through the details for the Hopf algebra k〈x _{1},x _{2},…,x _{ N }〉 and riffle shuffling (Examples 1.1 and 1.3). Section 5.1 gives background on shuffling, Section 5.2 develops the Hopf connection, Section 5.3 gives various descriptions of right eigenfunctions. These are specialized to decks with distinct cards in Section 5.3.1 which shows that the number of descents\(\frac {n1}{2}\) and the number of peaks\(\frac{n2}{3}\) are eigenfunctions. The last section treats decks with general composition showing that all eigenvalues 1/2^{ i }, 0≤i≤n−1, occur as long as there are at least two types of cards. Inverse riffle shuffling is a special case of walks on the chambers of a hyperplane arrangement and of random walks on a left regular band. Saliola [74] and Denham [25] give a description of left eigenfunctions (hence right eigenfunctions for forward shuffling) in this generality.
5.1 Riffle shuffles
The study of Q _{ a }(σ) has contacts with other areas of mathematics: to Solomon’s descent algebra [32, 83], quasisymmetric functions [36, 86, 87], hyperplane arrangements [6, 13, 16], Lie theory [69, 70], and, as the present paper shows, Hopf algebras. A survey of this and other connections is in [26] with [5, 22, 33] bringing this up to date. A good elementary textbook treatment is in [43].
It is also of interest to study decks with repeated cards. For example, if suits do not matter, the deck may be regarded as having values 1,2,…,13 with value i repeated four times. Now, mixing requires fewer shuffles; see [5, 23, 24] for details. The present Hopf analysis works here, too.
5.2 The Hopf connection
Let \(\mathcal{H}=k\langle x_{1},x_{2},\ldots,x_{N}\rangle\) be the free associative algebra on N generators, with each x _{ i } primitive. As explained in Examples 1.1 and 1.3, the map x→Ψ ^{ a }(x)/a ^{degx } is exactly inverse ashuffling. Observe that the number of cards having each value is unchanged during shuffling. This naturally leads to the following finer grading on the free associative algebra: for each \(\nu=(\nu_{1},\nu_{2}, \ldots,\nu_{N})\in\mathbb{N}^{N}\), define \(\mathcal{H}_{\nu}\) to be the subspace spanned by words where x _{ i } appears ν _{ i } times. The ath power map \(\varPsi^{a}=m^{[a]}\Delta^{[a]}:\mathcal{H}\to \mathcal{H}\) preserves this finer grading. The subspace \(\mathcal {H}_{1^{N}}\subseteq k\langle x_{1},x_{2},\ldots,x_{N}\rangle\) is spanned by words of degree 1 in each variable. A basis is \(\{x_{\sigma}=x_{\sigma ^{1}(1)}\cdots x_{\sigma^{1}(n)}\}\). The mapping Ψ ^{ a } preserves \(\mathcal{H}_{1^{n}}\) and \(\frac{1}{a^{n}}\varPsi^{a}(x_{\sigma})=\sum_{\pi}Q_{a}(\pi\sigma^{1})x_{\pi}\). With obvious modification the same result holds for any subspace \(\mathcal{H}_{\nu}\). Working on the dual space \(\mathcal{H}^{*}\) gives the usual Gilbert–Shannon–Reeds riffle shuffles. Let us record this formally; say that a deck has composition ν if there are ν _{ i } cards of value i.
Proposition 5.1
Let ν=(ν _{1},ν _{2},…,ν _{ N }) be a composition of n. For any a∈{1,2,…}, the mapping \(\frac{1}{a^{n}}\varPsi^{a}\) preserves \(\mathcal{H}_{\nu}\) and the matrix of this map in the monomial basis is the transpose of the transition matrix for the inverse ashuffling Markov chain for a deck of composition ν. The dual mapping is the Gilbert–Shannon–Reeds measure (5.1) on decks with this composition.
Note
Since the cards behave equally independent of their labels, any particular deck of interest can be relabeled so that ν _{1}≥ν _{2}≥⋯≥ν _{ N }. In other words, it suffices to work with \(\mathcal{H}_{\nu}\) for partitions ν.
5.3 Right eigenfunctions
Theorem 3.16 applied to the free associative algebra gives a basis of left eigenfunctions for inverse shuffles, which are right eigenfunctions for the forward GSR riffle shuffles. By Remark 2 after Theorem 3.16, each word \(w\in\mathcal{H}_{\nu}\) corresponds to a right eigenfunction f _{ w } for the GSR measure on decks with composition ν. As explained in Example 1.3 and Sect. 2.3, these are formed by factoring w into Lyndon words, standard bracketing each Lyndon factor, then expanding and summing over the symmetrization. The eigenvalue is a ^{ k−n } with k the number of Lyndon factors of w. The following examples should help understanding.
Example 5.2
For n=3, with ν=1^{3}, \(\mathcal{H}_{\nu}\) is sixdimensional with basis \(\{x_{\sigma}\}_{\sigma\in S_{3}}\). Consider w=x _{1} x _{2} x _{3}. This is a Lyndon word so no symmetrization is needed. The standard bracketing λ(x _{1} x _{2} x _{3})=[λ(x _{1}),λ(x _{2} x _{3})]=[x _{1},[x _{2},x _{3}]]=x _{1} x _{2} x _{3}−x _{1} x _{3} x _{2}−x _{2} x _{3} x _{1}+x _{3} x _{2} x _{1}. With the labeling of the transition matrix K _{ a } of (5.1), the associated eigenvector is (1,1,0,−1,0,−1)^{ T } with eigenvalue 1/a ^{2}.
For n=3,ν=(2,1), \(\mathcal{H}_{\nu}\) is threedimensional with basis \(\{x_{1}^{2}x_{2},x_{1}x_{2}x_{1},x_{2}x_{1}^{2}\}\). Consider \(w=x_{2}x_{1}^{2}\). This factors into Lyndon words as x _{2}⋅x _{1}⋅x _{1}; symmetrizing gives the eigenvector \(x_{1}^{2}x_{2}+x_{1}x_{2}x_{1}+x_{2}x_{1}^{2}\) or (1,1,1)^{ T } with eigenvalue 1.
The description of right eigenvectors can be made more explicit. This is carried forward in the following two sections.
5.3.1 Right eigenfunctions for decks with distinct values
Recall from Sect. 2.3 that the values of an eigenfunction f _{ w } at w′ can be calculated graphically from the decreasing Lyndon hedgerow T _{ w } of w. When w has all letters distinct, this calculation simplifies neatly. To state this, extend the definition of f _{ l } (l a Lyndon word with distinct letters) to words longer than l, also with distinct letters: f _{ l }(w) is f _{ l } evaluated on the subword of w whose letters are those of l, if such a subword exists, and 0 otherwise. (Here, a subword always consists of consecutive letters of the original word.) Because w has distinct letters, there is at most one such subword. For example, f _{35}(14253)=f _{35}(53)=−1.
Proposition 5.3
Let w be a word with distinct letters and Lyndon factorization l _{1} l _{2}⋯l _{ k }. Then, for all w′ with distinct letters and of the same length as w, \(f_{w}(w')=f_{l_{1}}(w')f_{l_{2}}(w')\cdots f_{l_{k}}(w')\) and f _{ w } takes only values 1, −1 and 0.
Example 5.4
f _{35142}(14253)=f _{35}(14253)f _{142}(14253)=−1⋅1=−1 as calculated in Sect. 2.3.
Proof
Recall from Sect. 2.3 that f _{ w }(w′) is the signed number of ways to permute the branches and trees of T _{ w } to spell w′. When w and w′ each consist of distinct letters, such a permutation, if it exists, is unique. This gives the second assertion of the proposition. This permutation is precisely given by permuting the branches of each \(T_{l_{i}}\) so they spell subwords of w′. The total number of branch permutations is the sum of the number of branch permutations of each \(T_{l_{i}}\). Taking parity of this statement gives the first assertion of the proposition. □
The example above employed a further shortcut that is worth pointing out: the Lyndon factors of a word with distinct letters start precisely at the record minima (working left to right, thus 35142 has minima 3,1 in positions 1,3), since a word with distinct letters is Lyndon if and only if its first letter is minimal. This leads to
Proposition 5.5
The multiplicity of the eigenvalue a ^{ n−k } on \(\mathcal {H}_{1,1,\ldots, 1}\) is c(n,k), the signless Stirling number of the first kind.
Proof
By the above observation, the multiplicity of the eigenvalue a ^{ n−k } on \(\mathcal{H}_{1,1, \ldots, 1}\) is the number of permutations with k record minima, which is also the number of permutations with k cycles, by [85, Prop. 1.3.1]. □
Example 5.6
(Invariance under reversal)
Let \(\bar{w}\) denote the reverse of w, then, for any σ, switching branches at every node shows that \(f_{\sigma}(\bar{w})=\pm f_{\sigma}(w)\) where the parity is n−# Lyndon factors in σ. Thus, each eigenspace of Ψ ^{ a } is invariant under the map \(w\to \bar{w}\). For example, f _{35142}(35241)=f _{35}(35241)f _{142}(35241)=1⋅1=−f _{35142}(14253) when compared with Example 5.4 above. The quantity (n−# Lyndon factors in 35142) is 5−2=3, hence the change in sign.
Example 5.7
(Eigenvalue 1)
σ=n,n−1,…,1 is the only word with n Lyndon factors, so f _{ σ }(w)=1 spans the 1eigenspace.
Example 5.8
(Eigenvalue 1/a and descents)
Example 5.9
(Eigenvalue 1/a ^{2} and peaks)
 Case 1

after l with l≠i, l≠j,l<k (e.g., 42513);
 Case 2

after j, i.e., σ=n,n−1,…,k+1,k−1,…,j+1,j−1,…,i+1,i,j,k,i−1,…,1 (e.g., 42135);
 Case 3

after i, i.e., σ=n,n−1,…,k+1,k−1,…,j+1,j−1,…,i+1,i,k,j,i−1,…,1 (e.g., 42153).
 Case 1

1 if ij and kl both occur as subwords of w, or if ji and lk both occur; −1 if ji and kl both occur, or if ij and lk both occur; 0 if i is not adjacent to j in w or if k is not adjacent to l (this is f _{ ij } f _{ lk });
 Case 2

1 if ijk or kji occur as subwords of w; −1 if ikj or jki occur; 0 otherwise (this is f _{ ij } f _{ jk }+f _{ ik } f _{ jk });
 Case 3

1 if ikj or jki occur as subwords of w; −1 if kij or jik occur; 0 otherwise (this is −f _{ ik } f _{ jk }+f _{ ij } f _{ ik }).
Proposition 5.10
f _{∧}(w):= # peaks in \(w\frac{n2}{3}\), f _{∨}(w):= # troughs in \(w\frac{n2}{3}\) and f _{−}(w):= # straights in \(w\frac{n2}{3}\) are right eigenfunctions with eigenvalue 1/a ^{2}.
Proof
It may be possible to continue the analysis to patterns of longer length; in particular, one interesting open question is which linear combinations of patterns and constant functions give eigenfunctions.
5.3.2 Right eigenfunctions for decks with general composition
Recall from Proposition 5.1 that, for a composition ν=(ν _{1},ν _{2},…,ν _{ N }) of n, the map \(\frac{1}{a^{n}}\varPsi ^{a}\) describes inverse ashuffling for a deck of composition ν, i.e., a deck of n cards where ν _{ i } cards have value i. Theorem 3.16 applies here to determine a full left eigenbasis (i.e., a right eigenbasis for forward shuffles). The special case of ν=(n−1,1) (follow one labeled card) is worked out in [21] and used to bound the expected number of correct guesses in feedback experiments. His work shows that the same set of eigenvalues {1,1/a,1/a ^{2},…,1/a ^{ n−1}} occur.
This section shows that this is true for all deck compositions (provided N>1). It also determines a basis of eigenfunctions with eigenvalue 1/a and constructs an eigenfunction which depends only on an appropriately defined number of descents, akin to Example 5.8.
The following proposition finds one “easy” eigenfunction for each eigenvalue of 1/a ^{ k }. The examples that follow the proof show again that eigenfunctions can correspond to natural observables.
Proposition 5.11
Fix a composition ν of n. The dimension of the 1/a ^{ k }eigenspace for the ashuffles of a deck of composition ν is bounded below by the number of Lyndon words in the alphabet {1,2,…,N} of length k+1 in which letter i occurs at most ν _{ i } times. In particular, 1/a ^{ k } does occur as an eigenvalue for each k, 0≤k≤n−1.
Proof
By remark 2 after Theorem 3.16, the dimension of the 1/a ^{ k }eigenspace is the number of monomials in \(\mathcal{H}_{\nu }\) with n−k Lyndon factors. One way of constructing such monomials is to choose a Lyndon word of length k+1 in which letter i occurs at most ν _{ i } times, and leave the remaining n−k letters of ν as singleton Lyndon factors. The monomial is then obtained by putting these factors in decreasing order. This shows the lower bound.
To see that 1/a ^{ k } is an eigenvalue for all k, it suffices to construct, for each k, a Lyndon word of length k+1 in which letter i occurs at most ν _{ i } times. For k>ν _{1}, this may be achieved by placing the smallest k+1 values in increasing order. For k≤ν _{1}, take the word with k 1s followed by a 2. □
Example 5.12
Example 5.13
For an ncard deck of composition ν, the second largest eigenvalue is 1/a. Our choice of eigenvectors correspond to words with n−1 Lyndon factors. Each such word must have n−2 singleton Lyndon factors and a Lyndon factor of length 2. Hence the bound in Proposition 5.11 is attained; furthermore it can be explicitly calculated: the Lyndon words of length 2 are precisely a lower value followed by a higher value, so the multiplicity of eigenvalue 1/a is \(\binom{N}{2}\). This does not depend on ν, only on the number N of distinct values.
Summing these eigenfunctions and arguing as in Example 5.8 gives
Proposition 5.14
For any ncard deck of composition ν, let a(w), d(w) be the number of strict ascents, descents in w, respectively. Then a(w)−d(w) is an eigenfunction of Ψ ^{ a } with eigenvalue 1/a.
Proof
Remarks
Under the uniform distribution, the expectation of a(w)−d(w) is zero. If initially the deck is arranged in increasing order w ^{0}, a(w ^{0})−d(w ^{0})=N−1. If w ^{ k } is the permutation after k ashuffles, the proposition gives \(E\{a(w^{k})d(w^{k})\}=\frac{1}{a^{k}}(N1)\). Thus for a=2, k=log_{2}(N−1)+θ shuffles suffice to make this expected value 2^{−θ }. On the other hand, consider a deck with n cards labeled 1 and n cards labeled 2. If the initial order is w ^{0}=11⋯12⋯21, a(w ^{0})−d(w ^{0})=0 and so E{a(w ^{ k })−d(w ^{ k })}=0 for all k.
Central limit theorems for the distribution of descents in permutations of multisets are developed in [24].
Example 5.15
6 Examples and counterexamples
This section contains a collection of examples where either the Hopfsquare map leads to a Markov chain with a reasonable “real world” interpretation—Markov chains on simplicial complexes and quantum groups—or the constructions do not work out to give Markov chains—a quotient of the symmetric functions algebra, Sweedler’s Hopf algebra and the Steenrod algebra. Further examples will be developed in depth in future work.
Example 6.1
(A Markov chain on simplicial complexes)
The associated Markov chain, restricted to complexes on n vertices, is simple to describe: from a given complex \(\mathcal{C}\), color the vertices red or blue, independently, with probability 1/2. Take the disjoint union of the complex induced by the red vertices with the complex induced by the blue vertices. As usual, the process terminates at the trivial complex consisting of n isolated vertices.
This Markov chain is of interest in quantifying the results of a “topological statistics” analysis. There, a data set (n points in a metric space) gives rise to a family of complexes \(\mathcal {C}_{\epsilon}\), 0≤ϵ<∞, where the vertices of each \(\mathcal {C}_{\epsilon}\) are the data points, and k points form a simplex if the intersection of the ϵ balls around each point is nonempty in the ambient space. For ϵ small, the complex is trivial. For ϵ sufficiently large, the complex is the nsimplex. In topological statistics [17, 18] one studies things like the Betti numbers of \(\mathcal{C}_{\epsilon}\) as a function of ϵ. If these are stable for a range of ϵ this indicates interpretable structure in the data.
Consider now a data set with large n and ϵ fixed. If a random subset of k points is considered (a frequent computational ploy) the induced subcomplex essentially has the distribution of the “painted red” subcomplex (if the painting is done with probability k/n). Iterating the Markov chain corresponds to taking smaller samples.
If the Markov chain starts out at the nsimplex, every connected subset of the resulting Markov chain is a simplex. Thus, at each stage, all of the higher Betti numbers are zero and β _{0} after k steps is 2^{ k }−X _{ k }, where X _{ k } is distributed as the number of empty cells if n balls are dropped into 2^{ k } boxes. This is a thoroughly studied problem [53]. The distribution of the Betti numbers for more interesting starting complexes is a novel, challenging problem. Indeed, consider the triangulation of the torus with 2n ^{2} initial triangles. Coloring the vertices red or blue with probability 1/2, the edges with red/red vertices are distributed in the same way as the “open sites” in site percolation on a triangular lattice. Computing the Betti number β _{0} amounts to computing the number of connected components in site percolation. In the infinite triangular lattice, it is known that p=1/2 is the critical threshold and at criticality, the chance that the component containing the origin has size greater than k falls off as k ^{−5/48}. These and related facts about site percolation are among the deepest results in modern probability. See [42, 81, 93] for background and recent results. Iterates of the Markov chain result in site percolation with p below the critical value but estimating β _{0} is still challenging.
It is natural to study the absorption of this chain started at the initial complex \(\mathcal{C}\). This can be studied using the results of Sect. 3.7.
Proposition 6.2
Let the simplicial complex Markov chain start at the complex \(\mathcal{C}\). Let G be the graph of the 1skeleton of \(\mathcal{C}\). Suppose that the chromatic polynomial of G is p(x). Then the probability of absorption after k steps is p(2^{ k })/2^{ nk } (with \(n=\mathcal{X}\)).
For example, if \(\mathcal{C}\) is the nsimplex, p(x)=x(x−1)⋯(x−n+1) and \(P\{\mbox{absorption after } k \mbox{ steps}\}=\prod_{i=1}^{n1}(1i/2^{k})\sim e^{2^{(2c1)}}\) if k=2(log_{2} n+c) for n large. If \(\mathcal{C}\) is a tree, p(x)=x(x−1)^{ n−1} and \(P\{\mbox{absorption after } k \mbox{ steps}\}=(11/2^{k})^{n1}\sim e^{2^{c}}\) if k=log_{2} n+c for n large.
Using results on the birthday problem in nonstandard situations [8, 20] it is possible to do similar asymptotics for variables such as the number of lsimplices remaining after k steps for more interesting starting \(\mathcal{C}\) such as a triangulation of the torus into 2(n−1)^{2} triangles.
As a final remark, note that the simplicial complex Markov chain induces a Markov chain on the successive 1skeletons. The eigenvectors of this Markov chain are beautifully developed in [35]. These all lift to eigenvectors of the complex chain, so much is known.
Example 6.3
(Quantized shuffle algebras)
When q=1 this shuffle product gives Ree’s shuffle algebra and general values of q lead to elegant combinatorial formulations of quantum groups. For general q>0, there is also a naturally associated Markov chain. Work on the piece with multigrading 1^{ n }, so each variable appears once and we may work with permutations in S _{ n }. For a starting permutation π, and 0≤j≤n, let \(\theta(j)=\sum_{S=j}q^{\operatorname{wt}(S,\pi)}\). Set \(\theta=\sum_{j=0}^{n}\theta(j)\). Choose j with probability θ(j)/θ and then S (with S=j) with probability \(q^{\operatorname {wt}(S,\pi)}/\theta(j)\). Move to \(\pi_{S}\pi_{S^{\mathcal{C}}}\). This defines a Markov transition matrix K _{ q }(π,π′) via \(\frac{1}{\theta}m\Delta(\pi)\). Note that the normalization θ depends on π.
The preceding description gives inverse riffle shuffles. It is straightforward to describe the qanalog of forward riffle shuffles by taking inverses. Let [j]_{ q }=1+q+⋯+q ^{ j−1}, [j]_{ q }!=[j]_{ q }[j−1]_{ q }⋯[1]_{ q }, and Open image in new window , the usual qbinomial coefficient. We write I(w) for the number of inversions of the permutation w and R(w)=d(w ^{−1})+1 for the number of rising sequences.
Proposition 6.4
Proof
Equation (6.1) follows from the inverse description because I(w)=I(w ^{−1}). For the sequential description, it is classical that Open image in new window is the generating function for multisets containing j ones and n−j twos by number of inversion [85, Sect. 1.7]. For two piles with say, 1,2,…,j in the left and j+1,…,k in the right, in order, dropping j induces k−j inversions. Multiplying the factors in (6.2) results in a permutation with R(w)≤2, with probability Open image in new window . Since the cut is made with probability Open image in new window , the twostage procedure gives (6.1). □
Remarks
When q=1, this becomes the usual Gilbert–Shannon–Reeds measure described in the introduction and in Sect. 5. In particular, for the sequential version, the cut is j with probability \(\binom{n}{j}/2^{n}\) and with A in the left and B in the right, drop from left or right with probability \(\frac{A}{A+B}\), \(\frac{B}{A+B}\), respectively. For general q, as far as we know, there is no closed form for z _{ n }. As q→∞, the cutting distribution is peaked at n/2 and most cards are dropped from the left pile. The most likely permutation arises from cutting off n/2 cards and placing them at the bottom. As q→0, the cutting distribution tends to uniform on {0,1,…,n} and most cards are dropped from the right pile. The most likely permutation is the identity. There is a natural extension to a q–ashuffle with cards cut into a piles according to the qmultinomial distribution and cards dropped sequentially with probability “qproportional” to packet size.
We hope to analyze this Markov chain in future work. See [30] for related qdeformations of a familiar random walk.
Example 6.5
(A quotient of the algebra of symmetric functions)
Example 6.6
(Sweedler’s example)
Example 6.7
(The Steenrod algebra)
Steenrod squares (and higher powers) are a basic tool of algebraic topology [46]. They give rise to a Hopf algebra A _{2} over \(\mathbb{F}_{2}\). Its dual \(A_{2}^{*}\) is a commutative, noncommutative Hopf algebra over \(\mathbb{F}_{2}\) with a simple description. As an algebra, \(A_{2}^{*}=\mathbb{F}_{2}[x_{1},x_{2},\ldots]\) (polynomial algebra in countably many variables) graded with x _{ i } of degree 2^{ i }−1. The coproduct is \(\Delta(x_{n})=\sum_{i=0}^{n}x_{ni}^{2^{i}}\otimes x_{i}\) (x _{0}=1). Alas, because the coefficients are mod 2, we have been unable to find a probabilistic interpretation of mΔ. For example, \((A_{2}^{*})_{3}\) has basis \(\{x_{1}^{3},x_{2}\}\) and \(m\Delta(x_{1}^{3})=0\), \(m\Delta(x_{2})=x_{1}^{3}\) so (mΔ)^{2}≡0. Of course, high powers of operators can be of interest without positivity [31, 44].
Notes
Acknowledgements
P. Diaconis was supported in part by NSF grant DMS 0804324. C.Y. Amy Pang was supported in part by NSF grant DMS 0652817. A. Ram was supported in part by ARC grant DP0986774.
We thank Marcelo Aguiar, Federico Ardila, Nantel Bergeron, Megan Bernstein, Dan Bump, Gunner Carlsson, Ralph Cohen, David Hill, Peter J. McNamara, Susan Montgomery and Servando Pineda for their help and suggestions, and the anonymous reviewer for a wonderfully detailed review.
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