A plactic algebra action on bosonic particle configurations: The classical case

We study the plactic algebra and its action on bosonic particle configurations in the classical case. These particle configurations together with the action of the plactic generators can be identified with crystals of the quantum analogue of the symmetric tensor representations in type A. It turns out that this action factors over a quotient algebra that we call partic algebra, whose induced action on bosonic particle configurations is faithful. We describe a basis of the partic algebra explicitly in terms of a normal form for monomials, and we compute the center of the partic algebra.


Introduction
Let k be a field. Fix an integer N ≥ 3. The (local) plactic algebra P N is the unital associative k-algebra generated by a 1 , . . . , a N −1 subject to the plactic relations a i a i−1 a i = a i a i a i−1 for 2 ≤ i ≤ N − 1, (1) together with the commutativity relation a i a j = a j a i for |i − j| > 1.
Our results hold over an arbitrary unitary associative ring, if we adapt the notation, replacing vector spaces by free modules and so on. In statements about the center we need to assume commutativity of the ground ring. For simplicity we choose to work over a field. The plactic relations go back to Lascoux and Schützenberger [LS81]. They study the monoid defined by the "plaxic relations" (in the original, "plaxique" or "a placche") (1), (2) and the nonlocal Knuth relation, a slightly weaker commutativity relation (a i a j )a k = (a j a i )a k , a k (a i a j ) = a k (a j a i ) for i < k < j (in particular for |i − j| > 1). This monoid is isomorphic to the monoid of semistandard Young tableaux with entries 1, . . . , N − 1 (and multiplication defined by row bumping) by reading off the entries of a tableau from left to right and bottom to top, see [Ful97, Section 2.1] for the details.
The name local plactic algebra for the algebra defined by the relations (1), (2) and (3) goes back to [FG98] due to the additional "local" commutativity relation (3). Fomin and Greene develop a theory of Schur functions in noncommutative variables that applies in particular to the (local) plactic algebra, see [FG98, Example 2.6], including a generalized Littlewood-Richardson rule for Schur functions defined over the plactic algebra. The plactic algebra acts on Young diagrams by Schur operators, i.e. a i adds a box in the ith column if possible, and otherwise maps the diagram to zero [Fom95].
The monoid defined by the plactic relations (1), (2) and (3) appears as a Hall monoid or "quantic monoid" of type A N −1 in [Rei01], [Rei02]: Reineke  In particular, Reineke shows that for Q = A N −1 (with orientation given e.g. by i → (i − 1) for the vertices 2 ≤ i ≤ N − 1 of Q), the k-linearisation of the resulting monoid is isomorphic the plactic algebra as defined above, where the isomorphism classes of the one-dimensional simple modules [S i ] are mapped to the generators a i . This is furthermore identified with the positive half of the twisted quantum group at q = 0, which is obtained by twisting the multiplication and desymmetrizing the quantum Serre relations so that they can be rewritten without appearance of q −1 . In [Rei02] it is proven that the twisted (positive) half of the quantum group specialized to q = 0 is isomorphic to the linearisation of the Hall monoid. The desymmetrized Serre relations at q = 0 are the plactic relations. By Ringel's theorem [Rin90] we know that the positive half of the twisted quantum group is isomorphic to the generic Hall algebra for any Dynkin quiver Q. Hence, the specialisation of the generic Hall algebra at q = 0 gives the Hall monoid. Different normal forms for monomials in the plactic algebra are given in terms of enumerations of the roots [Rei02, Theorem 2.10].
In [KS10] the plactic algebra appears in the study of bosonic particle configurations. Schur functions in the generators of the affine plactic algebra are defined using Bethe Ansatz techniques to show that they are well-defined despite the noncommutativity of the generators. Combinatorially, a bosonic particle configuration is given by a tuple (k 1 , . . . , k N −1 , k 0 ) in Z N ≥0 . One can think of such a tuple as a finite number of particles distributed on a discrete lattice of N positions on a line segment (the classical case) or along a circle (the affine case). Here we focus on the classical case. The generators a i act on the particle configurations and their k-span by lowering k i by 1 and increasing k i+1 by 1, if possible. If not possible since k i = 0, the result is 0. In the picture this would correspond to propagation of a particle from position i to i + 1. One can identify bosonic particle configurations with Young diagrams, then the operator a i acts by adding a box in the (i + 1)-st row of the Young diagram, if possible, and by 0 otherwise. Up to an index shift and switching rows and columns, this is the same as the action on Young diagrams by Schur operators from [FG98]. We will use the identification of the k-span of bosonic particle configurations with the vector space of polynomials k[x 1 , . . . , x N −1 , x 0 ] so that a particle configuration (k 1 , . . . , k N −1 , k 0 ) corresponds to a monomial x k1 Then the generator a i of the plactic algebra acts by lowering the exponent of x i by 1 and raising the exponent of x i+1 by 1. Note that this action is combinatorial in the sense of [FG98]. A very prominent combinatorial realization of the action of the generators a i of the plactic algebra on bosonic particle configurations with k particles is the action of the Kashiwara operators f i on the crystal B(kω 1 ) of type A N −1 , i.e. the crystal for the quantum analogue of the symmetric representation L(kω 1 ) = Sym k (C N ) of the Lie algebra sl N (C) (see e.g. [HK02] for details). The Young tableaux that constitute the nodes in the crystal graph correspond to the particle configurations by placing a particle at position i for each box labelled i in the Young tableau. Similarly, the crystal B(ω k ) for the quantum analogue of the alternating representation can be identified with fermionic particle configurations together with the action of the generators of the plactic algebra.
This implies that the Kashiwara operators satisfy the plactic relations on B(ω k ) and B(kω 1 ), see also [Mei16,Chapter I.1]. Relations among Kashiwara operators for abstract crystals of simply laced finite and affine type were studied by Stembridge in [Ste03] where a list of relations is given that hold if and only if the abstract crystal graph can be realized as a crystal graph of an integrable highest weight representation.
Here we study the representation of the plactic algebra on bosonic particle configurations more closely. Our main goal is to identify the kernel of this representation and to describe the resulting algebra. For Young diagrams our results can be interpreted as a full list of generating relations among the Schur operators. For crystals they can be interpreted as relations satisfied by the Kashiwara operators f i on crystals of the form B(kω 1 ).
Let us point out that from [KS10, Proposition 9.1], [BFZ96, Proposition 2.4.1], [BJS93] it is known that on fermionic particle configurations the nilTemperley-Lieb quotient of the plactic algebra acts faithfully. In [BM16] the case of affine fermionic particle configurations was studied, including a description of a normal form for monomials in the affine nilTemperley-Lieb algebra and its center.
Organization and Results. In Section 1 we introduce a quotient of the classical plactic algebra named partic algebra, and we start by some small technical preliminaries. In Section 2 we construct a normal form of the monomials in the partic algebra, leading to the Basis Theorem 2.1: Theorem. The partic algebra P part N has a basis given by monomials of the form In Section 3 we discuss the action of the classical plactic and the partic algebra on bosonic particle configurations which we realize as an action on the polynomial ring k[x 1 , . . . , x N −1 , x 0 ], and we prove faithfulness of the action of the partic algebra in Theorem 3.2: Theorem. The action of the partic algebra P part In Section 4 we describe the center of the partic algebra in terms of Theorem 4.1: Theorem. The center of the partic algebra P part N is given by the k-span of the elements In Section 5 we give an outlook to the much harder affine case treated in the followup work [Mei]. We recall the definition of the affine partic algebra and its action on affine bosonic particle configurations. For the description of the kernel we find an unexpected generalization of the partic relation from the classical case.

The partic algebra
We introduce a quotient of the classical plactic algebra: Definition 1.1. Define the partic algebra P part N to be the quotient of P N by the additional relations Note that one can interpret the plactic relations (1), (2) as commutativity of the product (a i+1 a i ) with the generators a i+1 and a i . Relation (4) together with (1) implies in particular that (a i+1 a i ) and (a i a i−1 ) commute.
Remark 1.2. This relation appears naturally in the study of bosonic particle configurations, see Section 3. In contrast, in the Hall monoid of finite type A N −1 one cannot expect [S i+1 * S i ] and Remark 1.3. We have two gradings on both the plactic and the partic algebra: 1. All relations preserve the length of monomials, hence P N and P part N can be equipped with a Z-grading by the length of monomials.
2. All relations preserve the number of different generators in a monomial, hence P N and P part N can be equipped with a Z N −1 -grading that assigns to the generator a i the degree e i , the i-th standard basis vector in Z N −1 . This is a refinement of the above length grading.
Lemma 1.4. In the plactic (and hence also in the partic) algebra, the following relations hold: Proof.
1. The second equation of Lemma 1.4.(i) follows from the first by the plactic relation (1).
2. This equality follows from the calculation

A basis of the partic algebra
In this section we formulate the following main theorem: Theorem 2.1. The partic algebra P part N has a basis given by monomials of the form Our approach is based on the observation that it suffices to construct a normal form for monomials to obtain a k-basis for an algebra given by generators and monomial relations. This follows from In this section we show that every monomial in the partic algebra is equivalent to a monomial of the form (7). In Section 3 we observe that these monomials act pairwise differently on the particle configuration module, and we conclude that they must have been distinct.
Proposition 2.3. Every monomial in the partic algebra P part N is equivalent to a monomial of the Proof. The proof works by induction on the length of monomials. If the length is equal to 1, we have a i = a ki i for k i = 1, and the condition from (7) is preserved. For the induction step our goal is to show that (7) is preserved. Since we can commute a i with all a j as long as j = i ± 1, we only need to consider In order to prove that this can be rewritten as in (8), we have to show that either we can pass a i through to the right hand side, increasing the exponent k i by one, or we leave it at the left hand side, increasing d i by one.
The inequality condition (7) is automatically satisfied if we increase one of the k's, so there is nothing to check. The equality (8) is obvious since we only apply the commutativity relation (3).
The inequality condition (7) is preserved since k i−1 ≥ 1, and again we only apply the commutativity relation (3).
The inequality condition (7) is preserved since d i−1 ≥ 1, and as before we only apply the commutativity relation (3).
Remark 2.6. Let us compare our normal form with the monomial bases of the plactic algebra from [Rei02]: The plactic algebra P N surjects onto the partic algebra P part N , mapping generators to generators and hence monomials to monomials. Given a monomial of the normal form from Proposition 2.3, finding the (finitely many) preimages of basis monomials in the plactic algebra amounts to solving a system of linear equations over the nonnegative integers, i.e. finding lattice points inside a polyhedron.

The action on bosonic particle configurations
In this section we discuss an action of the plactic algebra P N on the polynomial ring k[x 1 , . . . , x N −1 , x 0 ] in N variables. It was defined in [KS10,Proposition 5.8]. We recall the definition here: Let This defines an action of the plactic algebra which factors over the partic algebra: Proof. This can be verified by direct computation.
In this section our goal is the proof of the following main theorem:  N on k[x 1 , . . . , x N −1 , x 0 ] is faithful.
Definition 3.4. We introduce the shorthand notation i := (k 1 , . . . , k N −1 , k 0 One can think of the monomial x k1 1 . . . Now we investigate the action of the partic algebra on the particle configuration module. Proposition 3.5. Fix a monomial a dN−1 N −1 . . . a d3 3 a d2 2 a k1 1 a k2 2 a k3 3 . . . a kN−1 N −1 in the partic algebra satisfying condition (7). There is a unique particle configuration with the number of particles minimal, i.e. a monomial in k[x 1 , . . . , x N −1 , x 0 ] of minimal degree, so that the monomial acts nontrivially on it. This minimal particle configuration is given by Proof. First we show that a k1 1 a k2 2 a k3 3 . . . a N −1 annihilates any particle configuration (r 1 , r 2 , r 3 , . . . , r N −1 , r 0 ) with r i < k i for some i. We compute Together with condition (7) it follows that the action of a monomial of the form a dN−1 N −1 . . . a d3 3 a d2 2 a k1 1 a k2 2 a k3 3 . . . a kN−1 N −1 on a particle configuration (r 1 , r 2 , r 3 , . . . , r N −1 , r 0 ) is nontrivial iff r i ≥ k i for all i (recall that r 0 ≥ 0 = k 0 is automatically satisfied). This proves that i in is indeed the minimal particle configuration on which the monomial acts nontrivially. Now compute the image of i in under the action of the monomial: Plug in r i = k i for all i to see that This proves Proposition 3.5.
Proof (of Theorem 2.1). By Proposition 2.3 any monomial in the partic algebra is equivalent to one of the form (7). We have shown in Proposition 3.5 that the action on the particle configuration module distinguishes any two monomials of the form (7), hence (7) describes a normal form for the monomials in the partic algebra P part N , hence a basis of P part N . Now Theorem 3.2 follows as a corollary from Proposition 3.5: Proof (of Theorem 3.2). We have seen in Proposition 3.5 that the normal form monomials, hence the basis elements in P part N act linearly independent on the particle configurations. In other words, the action of P part N is faithful.
Remark 3.6. The faithfulness of the action of the algebra P part N on the particle configuration module motivates us to give P part N the name "partic" algebra.
By Proposition 2.3 and Proposition 3.5, we can identify each monomial in the partic algebra uniquely by the minimal particle configuration j ∈ Z N ≥0 on which it acts nontrivially and the output particle configuration i ∈ Z N ≥0 that one gets back from the action of the monomial on j. Hence the following is welldefined: and j = (k 1 , k 2 , k 3 . . . , k N −1 , 0). The number of particles |i| = |j| = i k i in i and j is the same.
This labelling is made so that a ij · v(j) = v(i) in the notation of Definition 3.4. where the latter is only defined for r i > 0.
With this notation we can rewrite Corollary 2.4 to obtain the following multiplication rule.

The center of the partic algebra
Now that we have a basis of the partic algebra with a convenient labelling at our disposal, the goal of this section is to describe the center of the partic algebra P part N .
Theorem 4.1. The center of the partic algebra P part N is given by the k-span of the elements The monomial a r N −1 a r N −2 . . . a r 2 a r 1 2 3 4 5 6 7 8 0 Figure 3: Example for N = 9: The action of the central element (a 8 a 7 a 6 a 5 a 4 a 3 a 2 a 1 ) 5 on the particle configuration (5, 0, 0, 0, 0, 0, 0, 0, 0).
Proof. Let z := i,j c ij a ij be an element in the center, where we label the monomial a ij by minimal input and output particle configurations as in Definition 3.7, with coefficients c ij ∈ k. Notice that a (0,...,0,r)(r,0,...,0) commutes with all a i by equation (6) from Lemma 1.4.(ii). We show that c ij = 0 for all j that contain some i = 1, and for all i that contain some i = 0.
Let i ≥ 2. First we prove that c ij = 0 for all j that contain a particle at position i. Since c ij a ij is central, it commutes in particular with a i−1 a i−2 . . . a 2 a 1 . Using Corollary 3.9 we calculate (a i−1 a i−2 . . . a 2 a 1 )a ij = a (i∪{i})(j∪{1}) , Therefore (a i−1 a i−2 . . . a 2 a 1 )a ij = a ij (a i−1 a i−2 . . . a 2 a 1 ) for i / ∈ j. This we use to deduce that we have (a i−1 a i−2 . . . a 2 a 1 )z = z(a i−1 a i−2 . . . a 2 a 1 ) if and only if (a i−1 a i−2 . . . a 2 a 1 ) which holds if and only if The latter is precisely the equality Observe on the other hand that for fixed i the set of monomials is linearly independent since the sets ((j \ {i}) ∪ {1}) are all distinct for distinct j. Next, we show by induction on the number k i of particles at position i in j that all coefficients c ij are zero for k i ≥ 1: For k i = 1, the set (j \ {i}) ∪ {1} does not contain any particle at position i any more. Hence the monomial a i((j\{i})∪{1}) cannot appear in the left sum in equation (13), and so its coefficient c ij must have been zero. For the induction step, assume that the coefficient c ij is zero for all a ij with at most k i particles at position i in the minimal input particle configuration j. Consider some a ij with k i + 1 particles at position i in j. So the set (j \ {i}) ∪ {1} contains k i particles at position i in j, and so the monomial a i((j\{i})∪{1}) cannot appear in the sum (13). Therefore we see that the coefficient c ij must have been zero.
We have shown that any central element in P part N is of the form where the particle configurations j are of the form (r, 0, . . . , 0), r ∈ Z ≥0 . We use the convention that i + 1 = 0 for i = N − 1 which matches our definition of the action of the partic algebra P part N on the bosonic particle configuration module. Notice that 0 is never contained in the minimal input particle configuration, so that for 1 ≤ i ≤ N − 1 we have that i + 1 / ∈ j for all c ij = 0. Now we use a similar induction argument to show that c ij = 0 for all i that contain a particle at position i = 0. So let 1 ≤ i ≤ N − 1. Using Corollary 3.9 we calculate that Since we have shown already that i + 1 / ∈ j, we know that a i z = za i is nothing but the equality This in turn is equivalent to the equality which can be rewritten as Again, we observe that the set of monomials {a ((i\{i})∪{i+1})j | i + 1 / ∈ j, i ∈ i} is linearly independent for fixed i.
By induction on the number k ′ i of particles at position i in i we see that all coefficients c ij are zero for k ′ i ≥ 1: For k ′ i = 1, the set (i \ {i}) ∪ {i + 1} does not contain any particle at position i any more. Hence the monomial a ((i\{i})∪{i+1})j cannot appear in the right sum in equation (14), and its coefficient c ij must have been zero. For the induction step we assume that the coefficients for all a ij with at most k ′ i particles at position i in the output particle configuration i are zero. Consider some a ij with k ′ i + 1 particles at position i in i. So the set (i \ {i}) ∪ {i + 1} contains k ′ i particles at position i in j, and the monomial a ((i\{i})∪{i+1})j cannot appear in the sum (13). Again we see that its coefficient c ij must have been zero.
We have deduced now that only those monomials labelled by minimal input particle configurations j = (r, 0, . . . , 0) and output particle configuration i = (0, . . . , 0, s) may have nonzero coefficients. Since the number of particles has to be the same in i and j, any central element is of the form The affine plactic algebra acts on the polynomial ring k[x 1 , . . . , x N , q] in N + 1 variables as follows: This representation is called the affine bosonic particle representation of the affine plactic algebra P N . Similar to the bosonic particle configurations in the classical case one can identify a monomial x k1 1 . . . x kN N with a particle configuration on a circle with N positions, with k i particles lying at position i. The indeterminate q protocols how often we apply a 0 to a particle configuration. where m, m ′ , k i+1 , k i+2 , . . . , k i−2 ∈ Z ≥0 are nonnegative integers. These relations can be seen as the proper affine version of the partic relations (1), (2) and (4), since we find in [Mei] that they generate the kernel of the action of P N on the affine bosonic particle representation k[x 1 , . . . , x N , q].