Separation of Variables and the Computation of Fourier Transforms on Finite Groups, II
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Abstract
We present a general diagrammatic approach to the construction of efficient algorithms for computing the Fourier transform of a function on a finite group. By extending work which connects Bratteli diagrams to the construction of Fast Fourier Transform algorithms we make explicit use of the path algebra connection to the construction of Gel’fand–Tsetlin bases and work in the setting of quivers. We relate this framework to the construction of a configuration space derived from a Bratteli diagram. In this setting the complexity of an algorithm for computing a Fourier transform reduces to the calculation of the dimension of the associated configuration space. Our methods give improved upper bounds for computing the Fourier transform for the general linear groups over finite fields, the classical Weyl groups, and homogeneous spaces of finite groups, while also recovering the best known algorithms for the symmetric group.
Keywords
Generalized Fourier transform Bratteli diagram Path algebra Computational complexityMathematics Subject Classification
43A30 20C15 65T501 Introduction
The CooleyTukey algorithm is undoubtedly the most famous of the FFTs. It is a divideandconquer algorithm whose basic idea was first recorded by Gauss in unpublished work (see e.g. [22] for a brief history of the algorithm). The key step is to rewrite the DFT on a cyclic group \(C_N\) as a linear combination of DFTs on \(C_n < C_N\) (for \(n\mid N\)). Iterating this step for a chain of subgroups of \(C_N\) yields algorithms more efficient than a direct matrix–vector multiplication.
This divideandconquer algorithm produces efficiencies by reducing the “big” problem to smaller subproblems that have common structure and in fact are themselves, smaller versions of the original, that can be efficiently combined to produce the required result. In this paper we continue a line of work that generalizes this approach to nonabelian groups [30, 32, 33, 37]. In this case the common subproblems are repeated occurrences of particular pieces of matrix multiplications (e.g., repeated block and thus elementbyelement multiplications) whose existence is ensured by working with very specific kinds of bases for the irreducible matrix representations (and associated matrix elements) enabled by choices of group factorizations. Thus, there is in a sense, “divideandconquer” going on in both the group and its dual.
The bases are encoded via paths in a Bratteli diagram attached to the group of interest, which in turn means that irreducible matrix elements correspond to pairs of paths in the diagram, which for a given group element may only be nonzero when of a particular form. I.e., the “repeated units” of our divideandconquer amount to certain subgraphs of a Bratteli diagram and efficiencies are gained by recognizing their multiple appearances in the corresponding calculation. This is the guts of the “separation of variables” (SOV) approach first introduced in [32] and then extended in [28] via a quiverbased formalism.
In this paper we finally take on the problem of laying a proper axiomatic and logical foundation for this approach and in so doing also produce improved algorithms for the important families of classical Weyl groups \(B_n\) and \(D_n\) and the general linear groups over finite fields \(GL_n(\mathbb {F}_q)\):
Theorem 1.1
\(C(B_n)\le n(2n1)\vert B_n\vert .\)
Theorem 1.2
\(C(D_n)\le \frac{n(13n11)}{2}\vert D_n\vert .\)
Theorem 1.3
\(C(Gl_n(\mathbb {F}_q))\le \left( \frac{4^nq^{n+1}q}{4q1}\right) \vert Gl_n(q)\vert .\)
Improvements for the complexity of Fourier transforms on related homogeneous spaces are also presented. For example, let \(B_n/B_{nk}\) denote the homogenous space of the Weyl group \(B_n\).
Theorem 1.4
\(C(B_n/B_{nk})\le k(4n2k1)\frac{\vert B_n\vert }{\vert B_{nk}\vert }.\)
Moreover, our results extend to chains of semisimple algebras rather than just chains of group algebras. This will be explored in subsequent work [35].
In Sect. 2 we outline the preliminaries needed for our results, including a discussion of the mainideas behind the SOV approach, necessarily recapitulated (in an abbreviated format) in order to make this paper as selfcontained as possible (although we acknowledge—given the title—the dependence on part I [32]). In Sect. 3 we present the improved SOV approach in detail, rewriting an iterated product in the path algebra as a sequence of bilinear maps on the newly defined “configuration spaces” (vector spaces of quiver morphisms). In Sect. 4 we give factorizations and counts to prove the specific group complexity results (Theorems 1.1,1.2,1.3) and also recover previously known methods for \(S_n\) [28]. The results in Sect. 4 depend on various important, but very technical details of the explicit computation of the configuration space dimensions. In order to bring the reader to the complexity results as quickly and directly as seems possible, we postpone the presentation of these details to Sects. 5 and 6.1. This includes generalizations of some results of Stanley on differential posets [41, 42] used to give explicit methods for finding these dimensions. This may be of independent interest. Some of the more laborious (but necessary) formalisms are collected in three short appendices.
2 Background
2.1 Fourier Transforms and the Group Algebra
The usual discrete Fourier transform of a finite data sequence may be viewed as a special case of Fourier transforms on finite groups, defined using group representations. Results here assume complex representations, unless spelled out otherwise, although most results go through more generally. For necessary background on the representation theory of finite groups we refer the reader to [40].
Definition 2.1
 (i)Let \(\rho \) be a matrix representation of G. Then the Fourier transform of \(\mathbf {f}\) at \(\mathbf {\rho }\), denoted \(\hat{f}(\rho )\), is the matrix sum$$\begin{aligned} \hat{f}(\rho )=\sum _{s\in G} f(s)\rho (s).\end{aligned}$$
 (ii)Let R be a set of matrix representations of G. Then the Fourier transform of \(\mathbf {f}\) on \(\mathbf {R}\) is the direct sum of Fourier transforms of f at the representations in R:$$\begin{aligned} \mathcal {F}_R(f)=\bigoplus _{\rho \in R} \hat{f}(\rho )\in \bigoplus _{\rho \in R} {\text {Mat}}_{\dim \rho }(\mathbb {C}). \end{aligned}$$
When we compute the Fourier transform for a complete set of inequivalent irreducible representations R of G we refer to the calculation as the computation of a Fourier transform on G (with respect to R).
Definition 2.2
 (ii)
Let \(+_G(R)\) (respectively, \(\times _G(R)\)) denote the minimum number of complex arithmetic additions (resp., multiplications) needed to compute the Fourier transform of f on R via a straightline program^{1} for an arbitrary complexvalued function f defined on G. The arithmetic complexity of a Fourier transform on R, denoted \(T_G(R)\), is given by \(\max {(+_G(R), \times _G(R))}\).
 (ii)The complexity of the group G, denoted C(G) is defined bywhere R varies over all complete sets of inequivalent irreducible matrix representations of G.$$\begin{aligned} C(G):=\min _R \{T_G(R)\}, \end{aligned}$$
 (iii)The reduced complexity, denoted \(t_G(R)\), is defined by$$\begin{aligned} t_G(R)=\frac{1}{\vert G\vert }T_G(R). \end{aligned}$$
Example 2.3
The classical DFT and FFT. For \(G=C_N\), the cyclic group of order N, the irreducible representations are 1dimensional and defined by \(\zeta _j\rightarrow \zeta _j^k,\) for \(\zeta _j=e^{2\pi ij/N}\) and \(k=0,\dots , N1\) and \(i=\sqrt{1}\). The corresponding Fourier transform on \(C_N\) is the usual discrete Fourier transform. Cooley and Tukey’s algorithm showed that for a “highly composite” integer N (an integer N that factors completely as a product of small prime numbers), \(C(G)\le O(N\log _2 N)\) [7].
A complete set R of inequivalent irreducible matrix representations of a group G determines a basis for \(\mathbb {C}[G]\) (via the irreducible matrix elements) and in this case the Fourier transform is an algebra isomorphism from \(\mathbb {C}[G]\) to a direct sum of matrix algebras. We recover f through the Fourier inversion formula.
Theorem 2.4
Lemma 2.5
2.2 Adapted Bases, Bratteli Diagrams, and Quivers
The fundamental idea of the SOV approach is a recasting of the CooleyTukey algorithm in terms of graded quivers, which is an elaboration of path algebras derived from Bratteli diagrams, which are motivated by the use of adapted or Gel’fand–Tsetlin bases for irreducible representations.
Definition 2.6
Given a group G with subgroup \(H\le G\), a complete set R of inequivalent irreducible matrix representations of G is \(\mathbf {H}\) adapted if there exists a complete set \(R_H\) of inequivalent irreducible matrix representations of H such that for all \(\rho \in R\), \(\rho \downarrow _H=\bigoplus \gamma _s\), for (not neccessarily distinct) representations \(\gamma _s\) in \(R_H\). The set R is adapted to the chain \(G=G_n> G_{n1}>\cdots > G_0\) if for each \(1\le i\le n\) there is a complete set \(R_i\) of inequivalent representations of \(G_i\) such that \(R_i\) is \(G_{i1}\)adapted and \(R_n=R\). A set of bases for the representation spaces that give rise to adapted representations is an adapted basis.
For the FFT results of this paper we assume the ability to construct adapted sets of representations. This requirement is not a limitation, as any set of representations is equivalent to an adapted set of representations. One such construction is outlined in [32].
Definition 2.7
A quiver Q is a directed multigraph with vertex set V(Q) and edge set E(Q). For an arrow (directed edge) \(e\in E(Q)\) from vertex \(\beta \) to vertex \(\alpha \), we call \(\alpha \) the target of e and \(\beta \) the source of e.
Let Q be a quiver. For each \(e\in E(Q)\), let t(e) denote the target of e and s(e) the source of e.
Definition 2.8
A quiver Q is graded if there is a function \(gr:V(Q)\rightarrow \mathbb {N}\) such that for each \(e\in E(Q)\), \(gr(t(e))>gr(s(e))\).
Example 2.9
Figure 1 is an example of a graded quiver. Each vertex v is labeled by its grading, gr(v).
Definition 2.10
 (i)
there is a unique vertex with grading 0, called the root,
 (ii)
if \(v\in V(Q)\) is not the root then v is the target of at least one arrow,
 (iii)
if \(v\in V(Q)\) does not have grading of maximum value then v is the source of at least one arrow,
 (iv)
for each \(e\in E(Q)\), \(gr(t(e))=1+gr(s(e))\).
Example 2.11
Note that the quiver of Fig. 1 is not a Bratteli diagram. However, a slight modification produces the Bratteli diagram of Fig. 2.
Consider a group algebra chain \(\mathbb {C}[G_n]> \mathbb {C}[G_{n1}]> \dots> \mathbb {C}[G_1]> \mathbb {C}[G_0] = \mathbb {C}\). To associate a Bratelli diagram to this chain we follow the language of [36]. Let \(\rho \) be an irreducible representation of \(G_{i}\), i.e., an irreducible \(\mathbb {C}[G_i]\)module. Upon restriction to \(G_{i1}\), \(\rho \downarrow _{G_{i1}}\) decomposes as a direct sum of irreducible \(\mathbb {C}[G_{i1}]\)modules. For \(\gamma \) an irreducible representation of \(G_{i1}\), let \(M(\rho ,\gamma )\) denote the multiplicity of \(\gamma \) in \(\rho \downarrow _{G_{i1}}\).
Definition 2.12
 (i)
The vertices of grading i are labeled by the (equivalence classes of) irreducible representations of \(G_i\);
 (ii)
A vertex labeled by an irreducible representation \(\gamma \) of \(G_{i1}\) is connected to a vertex labeled by an irreducible representation \(\rho \) of \(G_{i}\) by \(M(\rho ,\gamma )\) arrows.
Example 2.13
Figure 3 shows two examples of Bratteli diagrams, with the gradings listed at the top.
On the left of Fig. 3 we see the Bratteli diagram for a chain of group algebras for \(C_6\) while on the right we see the Bratteli diagram for a chain of group algebras for the symmetric group \(S_3\), viewing \(S_i\) as the subgroup of \(S_n\) that fixes the elements \(\{i+1,\dots ,n\}\). Note that we distinguish \(\mathbb {C}[S_1]\) from \(\mathbb {C} (= C[S_0])\) only so that vertices at level i correspond to representations of \(\mathbb {C}[S_i]\).
For the group algebra \(\mathbb {C}[C_N]\), irreducible representations are naturally indexed by the integers \(0,\dots ,N1\), while for \(\mathbb {C}[S_n]\), the irreducible representations are indexed by partitions of n (as determined by Young in [46]; see [24] for an introduction to the representation theory of \(S_n\)).
Both Bratteli diagrams of Fig. 3 are examples of multiplicityfree diagrams in that there is at most one edge from any vertex of grading i to any vertex of grading \(i+1\).
Definition 2.14
Let \(\mathcal {B}\) be a Bratteli diagram. The path algebra (at level i), denoted \(\mathbb {C}[\mathcal {B}_i]\), is the \(\mathbb {C}\)vector space with basis given by ordered pairs of paths of length i in \(\mathcal {B}\) which start at the root and end at the same vertex at level i.
Example 2.15
In the Bratteli diagram \(\mathcal {B}\) of Fig. 4 associated to the chain \(\mathbb {C}[S_3]>\mathbb {C}[ S_2] > \mathbb {C}[S_1]\ge \mathbb {C},\) let \(P_1,P_2,P_3,P_4\) be the paths from the root to level 3 in \(\mathcal {B}\), labeled from top to bottom. Then the path algebra \(\mathbb {C}[\mathcal {B}_3]\) has basis \(\{(P_1,P_1), (P_2,P_2), (P_2,P_3), (P_3,P_2), (P_3,P_3), (P_4,P_4)\}.\)
Note that for a vertex v, labeled by a representation \(\rho \), the dimension of \(\rho \) is given by the number of paths from the root to v. Moreover, each path corresponds to a subgroupequivariant embedding of \(\mathbb {C}\) into the representation space of \(\rho \) (for more details, see “Appendix 1”).
For a Bratteli diagram \(\mathcal {B}\) with highest grading n associated to a chain of group algebras, consider the associated chain of path algebras: \(\mathbb {C}[\mathcal {B}_n]>\mathbb {C}[\mathcal {B}_{n1}]>\cdots> \mathbb {C}[\mathcal {B}_1]>\mathbb {C}[\mathcal {B}_0]=\mathbb {C}.\) It is not too difficult to see that there exists an isomorphism between these algebra chains.
Lemma 2.16
Let \(\mathbb {C}[G]=\mathbb {C}[G_n]> \mathbb {C}[G_{n1}]> \dots> \mathbb {C}[G_1]> \mathbb {C}[G_0] = \mathbb {C}\) be a chain of group algebras with Bratteli diagram \(\mathcal {B}\). Then the chain of path algebras associated to \(\mathcal {B}\) is isomorphic to the group algebra chain.
For further explanation see “Appendix 1” and Section 2.3 of [19].
Remark 2.17
Quivers were first introduced by Gabriel in the study of modular representation theory [17]. Bratteli diagrams were first introduced to classify inductive limits of \(C^*\)algebras [3]. After Elliot’s use of Bratteli diagrams in the classification of AFalgebras [16], these ideas motivated a program to classify \(C^*\)algebras in terms of their Ktheory [39]. In terms of the representation theory of semisimple algebras, Bratteli diagrams have been used to explicitly construct complete sets of irreducible representations that are analogs of Young’s seminormal form in the symmetric group, and to describe restriction relations of representations [9, 20, 21, 26].
2.3 Gel’fand–Tsetlin Bases
The analogous concept in the path algebra of
adapted bases associated to a group algebra chain is a system of Gel’fand Tsetlin bases.
Definition 2.18
Let \(\mathcal {B}\) be the Bratteli diagram associated to a chain of group algebras. A system of Gel’fand–Tsetlin bases for \(\mathbf {\mathcal {B}}\) consists of a collection of bases for the representation spaces \(\{V_\alpha \vert \;\alpha \in V(\mathcal {B})\}\) of the representations corresponding to \(\alpha \) indexed by paths from the root to \(\alpha \), along with maps from the paths to the basis vectors; i.e., a set of basis vectors along with knowledge of the path corresponding to each vector.
Example 2.19
Let \(\mathcal {B}\) be the Bratteli diagram of Fig. 3 associated to the chain \(\mathbb {C}[S_3]>\mathbb {C}[ S_2] > \mathbb {C}[S_1]\ge \mathbb {C}\). Then for the paths \(P_1,P_2,P_3,P_4\) defined in Example 2.15, a basis \(\{w_{P_2},w_{P_3}\}\) for the twodimensional representation space Open image in new window is part of a system of Gel’fand–Tsetlin bases for \(\mathcal {B}\). Note that the entries of the matrix of this representation are indexed by pairs \(\{(w_{P_i},w_{P_j})\mid i,j=1,2\}\) and so correspond to basis elements of the path algebra \(\mathbb {C}[\mathcal {B}_3]\).
Systems of Gel’fand–Tsetlin bases were originally developed by Gel’fand and Tsetlin to calculate the matrix coefficients of compact groups [18]. Clausen was the first to apply them to the efficient computation of Fourier transforms on finite groups [6].
In Remark 7.5 of “Appendix 1”, we show systems of Gel’fand–Tsetlin bases for the chain of path algebras corresponding to a group algebra chain are equivalent to adapted bases for the chain of subgroups. The notion of an adapted basis coincides with that of a set, for each \(1\le i\le n\), of \(G_i\)equivariant maps between the representation spaces of representations in \(R_i\) and those in \(R_{i+1}\). For further details, see “Appendix 1”.
Gel’fand–Tsetlin bases provide a means to better understand the isomorphism of Lemma 2.16 between a chain of group algebras and the corresponding chain of path algebras. Since Gel’fand–Tsetlin bases are indexed by paths in \(\mathcal {B}\) and a basis for the path algebra \(\mathbb {C}[\mathcal {B}_n]\) consists of pairs of paths, we identify the group algebra \(\mathbb {C}[G]\) with its realization in coordinates relative to the Gel’fand–Tsetlin basis, indexed by pairs of paths of length n in \(\mathcal {B}\) that share the same endpoint. For \(s\in G\) let \(\tilde{s}:=\sum _{(P,Q)\in \mathbb {C}[\mathcal {B}_n]} [s]_{P,Q}(P,Q).\) These are the coordinates of s in the path algebra basis. Then Lemma 2.5 becomes
Lemma 2.20
Example 2.21
Young’s orthogonal form gives an example of a complete set of irreducible matrix representations for \(S_n\) adapted to the chain \(S_n>S_{n1}>\cdots >S_1\). Since restriction of representations from \(S_n\) to \(S_{n1}\) is multiplicityfree, the basis vectors of a system of Gel’fand–Tsetlin bases for the irreducible representations relative to this chain are determined up to scalar multiplies, and in the case of \(n=3\), the paths are the paths \(P_1, P_2, P_3, P_4\) of Example 2.15. In [28], Maslen gives an efficient algorithm for computation of the Fourier transform of a function on \(S_n\) by considering the computation of \(\sum \nolimits _{s\in S_n}f(s)s\) in the group algebra for \(S_n\) relative to this Gel’fand–Tsetlin basis.
3 The Separation of Variables Approach
Lemma 3.1
Proof
 1.
For each \(y\in \tilde{Y}\), factor \(yF_y=x_1\cdots x_m\) in such a way as to enable rearrangements allowing for \(\mathcal {F}_Y\) to be a recursively structured summation.
 2.
Each factor \(x_i\) will correspond to an element of the path algebra of a particular form, and thus a particular subgraph of the Bratteli diagram. These subgraphs can be given a vector space structure through an identification with a space of quiver morphisms.
 3.
By virtue of the vector space identification, the element multiplication \(x_ix_{i+1}\) becomes a bilinear map whose complexity can be calculated directly in terms of the dimension of the derived space of graph morphisms.
Example 3.2
Suppose \(y\in \mathbb {C}[G]\) factors as \(y=x_1x_2\) with \(x_i\in \mathbb {C}[G_{i+2}]\cap {\text {Centralizer}}\mathbb {C}[G_{i}]\). Express \(x_i\) in Gelf’and Tsetlin coordinates as \(\tilde{x}_i=\sum _{(P,Q)}[x_i]_{PQ}(P,Q)\). An application of Schur’s Lemma and standard facts about Gel’fand–Tsetlin bases show that \([x_i]_{P,Q}\) is 0 unless P and Q are paths in \(\mathcal {B}\) that agree from level \(i+1\) to level n, and from level 0 to level \(i1\), as in the quivers \(Q_i\) of the lefthand side of Fig. 6 (see also [31]). The product \(\tilde{x}_1\tilde{x}_2\) is indexed by any triple of paths resulting from gluing \(Q_2\) to \(Q_1\) obtained by identifying the “bottom” path of \(Q_1\) with the “top” path of \(Q_2\), but these triples must simultaneously maintain the structures of \(Q_1\) and \(Q_2\) (the quiver on the righthand side of Fig. 6). The complexity count is thus the careful counting of these compatible structures, which can be recast as the computation of the dimension of a space of quiver morphisms.
For products with more factors we iterate this gluing process. Example 3.6 below gives further details. The SOV approach consists of factoring \(\tilde{y}F_y\), forming the graph (akin to the righthand side of Fig. 6) and determining the subgraphs (like the lefthand side of Fig. 6) corresponding to each individual product.
Definition 3.3
In this setting (bilinear) group algebra multiplication is transformed into a bilinear map on products of associated spaces of quiver morphisms. Call this map \(*\). As the notation and details are more technical than illuminating, we defer the explicit definition of \(*\) and discussion of its properties to Sect. 5. However, even with deferring this we can present the algorithm. Keep in mind the identification of the group algebra and the path algebra.
 I.
Choose \(m\in \mathbb {N}\) and a subset \(X\subseteq (\mathbb {C}[\mathcal {B}_n])^m\) such that \(\vert X\vert =\vert Y\vert \) and for each \(y\in Y\) there exists \((x_1,\dots , x_m)\in X\) with \(\tilde{y}F_y=x_1\cdots x_m.\) Thus, X can be thought of as a choice of factorization into m elements (some of which may be the identity) of each term \(yF_y\).
 II.
For \(1\le i\le m\) let \(X_i\) be as in Definition 3.3. For \(\sigma \in S_m\), let \(w_i= x_{\sigma (i)}\). The bilinear map \(*\) is such that \(x_1\cdots x_m=(((w_1*w_2)*w_3)\cdots *w_m),\)
 III.
For \( 0\le i<m\), let \(W_i=\{(w_{i+1},\dots , w_m)\mid (x_1,\dots ,x_m)\in X\}\). Let \( W_m=\emptyset . \) Note that \(W_i\subseteq X_{\sigma (i+1)}\times \cdots \times X_{\sigma (m)}\).
 IV.Define a sequence of functions \(L_i\) recursively by:$$\begin{aligned} \begin{array}{ll} L_1(w_2,\dots ,w_m)=&{}\displaystyle \sum _{(w_1,w_2,\dots ,w_m)\in W_0} w_1, \\ L_2(w_{3},\dots ,w_m)=&{}\displaystyle \sum _{(w_2,w_{3},\dots , w_m)\in W_{1}} L_{1}(w_2,\dots ,w_m)*w_2.\\ L_i(w_{i+1},\dots ,w_m)=&{}\displaystyle \sum _{(w_i,w_{i+1},\dots , w_m)\in W_{i1}} (L_{i1}(w_i,\dots ,w_m)*w_i). \end{array} \end{aligned}$$
Theorem 3.5
Proof
Follows from II. and induction. \(\square \)
Example 3.6
For \(\sigma =(123)\), \(w_1*w_2*w_3=x_2*x_3*x_1\). The complexity of \(x_2*x_3\) is \(\#{\text {Hom}}(Q_2\cup Q_3;\mathcal {B})\), where \(Q_2\cup Q_3\) is as in Fig. 9, the subquiver of \(\mathcal {Q}\) corresponding to \(Q_2\) and \(Q_3\) (note that in Fig. 9 we show only the subquiver formed by the segments of \(Q_2\cup Q_3\) where not all three – top, bottom and the summed over middle – of the paths agree). The complexity of \((x_2*x_3)*x_1\) is \(\#{\text {Hom}}((Q_2\triangle Q_3)\cup Q_1;\mathcal {B})\), where \(Q_2\triangle Q_3\) is the quiver of Fig. 9 associated to the space containing \(x_2*x_3\). Note that as per the notation \(Q_2\triangle Q_3\) is in fact the symmetric difference of \(Q_2\) and \(Q_3\), i.e., the edges of \(Q_2\cup Q_3\) not in \(Q_2\cap Q_3\) (see Definition 5.6).
Lemma 3.7
For \(Q_i\) (respectively, \(Q_j\)) the quiver associated to \(X_i\) (respectively, \(X_j\)), computation of \(x_i*x_j\) requires at most \(\#{\text {Hom}}(Q_i\cup Q_j;\mathcal {B})\) scalar multiplications and fewer additions.
We postpone the proof of this key counting lemma to Sect. 5. With Lemma 3.7 we now have our main general result:
Theorem 3.8
Proof

Stage 0: Find \(W_0\) by reordering X.

Stage 1: Compute \(L_1\) for all \((w_2,\dots ,w_m)\) in \(W_1\).

Stage i: Compute \(L_i\) given \(W_{i1}\) and \(L_{i1}\).
For an explicit additions count, see Theorem 5.14 in Sect. 5.
4 The Complexity of Fourier Transforms on Finite Groups
The SOV approach computes path algebra sums by first factoring each element and then translating multiplication into maps indexed by subgraphs. The complexity is determined by the size of the factorization sets and the number of occurrences of these subgraphs in the Bratteli diagram. Thus, our main results require methods to determine these counts. In this section we demonstrate the subgraphs determined by the SOV apporach and defer the proofs of the complexity counts to Sect. 6.1 and the appendices. In this way we hope to give the visual sense (and attendant justification of the proofs) of the algorithm without an overload of technical notation.
4.1 The Weyl Groups \(B_n\) and \(D_n\)
For our first application of the SOV approach we consider the Fourier transform of functions on the Weyl groups of type \(B_n\) and \(D_n\). We improve upon the results of [32].
Theorem 4.1
Proof
Let \(s_1,\dots ,s_n\) denote the simple reflections for \(B_n\), labeled as per the usual Dynkin diagram schema (see e.g., [23]) in Fig. 10.
 I.
Let \(X=\{(a_n,\dots ,a_2,a_1,a_2',\dots ,a_n', F_{a_n\cdots a_2a_1a_2'\cdots a_n'})\vert \; a_i\in \tilde{A}_i,a_i'\in \tilde{A}_i'\}.\)
 II.Note thatFigure 12 shows the various component subquivers corresponding to the coset representatives. They combine together as per Fig. 12 to give the factorization of \(yF_y\). Thus, the algorithm proceeds by gluing together quivers \(Q_{i}\) of Fig. 11 (corresponding to elements of \(\tilde{A}_j, \tilde{A}'_j, \text { or } F_y\), as per necessary) to build the quiver \(\mathcal {Q}\) of Fig. 12. The left column of Fig. 11 shows the quivers \(Q_{i}\) for \(1\le i\le n\) and the right column shows the quivers \(Q_{i}\) for \(n+1\le i\le 2n\).$$\begin{aligned} \begin{array}{ll} \displaystyle i^+=ni+1,&{} \quad 1\le i\le n,\\ \displaystyle i^+=in+1, &{}\quad n<i<2n,\\ i^=i^+2,&{}\quad 1\le i< 2n,\\ {2n}^+=n1,&{}\\ {2n}^=0.&{} \end{array} \end{aligned}$$
 III.Let \(\sigma \in S_{2n}\) be the permutation reordering X so that \(W_0\) is the set \(\{(F_{a_n\cdots a_2a_1a_2\cdots a_n}, a_2',a_3',\dots a_n',a_1,a_2,a_3,\dots ,a_n)\}.\) ThenNote that$$\begin{aligned} \begin{array}{l} W_1=\{(a_2',a_3',\dots a_n',a_1,a_2,a_3,\dots ,a_n)\vert \; a_i\in \tilde{A}_i,a_i'\in \tilde{A}_i'\},\\ W_2=\{(a_3',\dots a_n',a_1,a_2,a_3,\dots ,a_n)\vert \; a_i\in \tilde{A}_i, a_i'\in \tilde{A}_i'\},\\ \vdots \\ W_{2n1}=\{(a_n)\vert \;a_n\in \tilde{A}_n\}.\end{array} \end{aligned}$$$$\begin{aligned} \begin{array}{ll}\vert W_{i1}\vert =\vert \tilde{A}_i'\vert \cdots \vert \tilde{A}_n'\vert \vert \tilde{A}_1\vert \cdots \vert \tilde{A}_n\vert , &{} \quad 2\le i\le n,\\ \vert W_{i1}\vert =\vert \tilde{A}_{in}\vert \cdots \vert \tilde{A}_n\vert ,&{}\quad n<i\le 2n.\end{array} \end{aligned}$$
Analogous arguments give the following result for Weyl groups of type \(D_n\).
Theorem 4.2
Proof
Let \(s_1,\dots ,s_n\) denote the simple reflections for \(D_n\), labeled according to its standard Dynkin diagram (see Fig. 15).
4.2 The General Linear Group
Theorem 4.3
Proof
 I.
Let \(X=\{(u_2,\dots ,u_{p1},u_{p+1}',t_p,u_{p+1},\dots ,u_i,v_{i+1},\dots ,v_n, F_{u_2\cdots v_n})\}\), ranging over \(u_j\in \tilde{U}_j,u_j'\in \tilde{U}_j',v_j\in \tilde{V}_j,t_j\in \tilde{T}_j\) and \(i\le n\) with \(i\mid (p1)\}.\)
 II.
Fig. 16 shows the various component subquivers corresponding to the coset representatives. They combine together as per Fig. 17 to give the factorization of \(yF_y\). Thus, the algorithm proceeds by gluing together quivers \(Q_{i}\) of Fig. 16 to build the quiver \(\mathcal {Q}\) of Fig. 17.
 III.Let \(\sigma \in S_{n+m1}\) be the permutation reordering X so thatThen$$\begin{aligned} \begin{array}{l}W_0=\{( F_{u_2\cdots v_n}, u_2, u_3,\dots , u_{p1}, t_p, u_{p+1}, u_{p+1}', u_{p+2},...,u_{i},v_{i+1},\dots ,v_n)\}.\end{array} \end{aligned}$$$$\begin{aligned} \begin{array}{l} W_1=\{(u_2, u_3,\dots , u_{p1}, t_p, u_{p+1}, u_{p+1}', u_{p+2},...,u_{i},v_{i+1},\dots ,v_n)\}\\ W_2=\{(u_3,\dots , u_{p1}, t_p, u_{p+1}, u_{p+1}', u_{p+2},...,u_{i},v_{i+1},\dots ,v_n)\}\\ \vdots \\ W_{m+n2}=\{( v_n)\}. \end{array} \end{aligned}$$
4.3 Generalized Symmetric Group Case
We next give a general result (Theorem 4.4) to find efficient Fourier transforms on groups with special subgroup structure. As the proof follows the same structure of the proofs of Theorems 10, 15, and 1.3, we leave it as an exercise.
 (1)
\(A_1=G_1\),
 (2)
\( G_i=A_2\cdots A_i G_{i1}\) for \(2\le i\le n\),
 (3)
\(A_i\) commutes with \(G_{i2}.\)
Theorem 4.4
Note 4.5
Theorem 4.4 is a refinement of Theorem 3.1 of [30]: rather than considering the maximum length of a factorization in terms of coset representatives, we need only multiply by \(\prod \vert A_j\vert \). Note that our choice of coset representatives in the proofs of Theorems 1.1 and 1.2 were such that \(\prod \vert A_j\vert =1\), much smaller than the length of the longest factorization in terms of coset representatives.
Note 4.6
For \(G_i=S_i\), this theorem gives an efficient algorithm for the computation of the Fourier transform of a function on the symmetric group by letting \(A_1=\{e\}\) and \(A_i=\{e, t_{i1}\}\) for \(2\le i\le n\).
4.4 The Complexity of Fourier Transforms on Homogeneous Spaces
We next consider the Fourier transform of a function on a homogeneous space, a special case of harmonic analysis on groups. This can be viewed as a coset space G / K, so a Fourier transform on a homogeneous space is a Fourier transform of the space of functions on G / K or, equivalently, of the space of associated rightK invariant functions on G. See [28, 32] for further background on Fourier transforms on homogeneous spaces and some of their applications.
Definition 4.7
Note that \(\hat{f}(\rho )\) is zero unless the representation space, \(V_\rho \), contains a nontrivial Kinvariant vector. Such a representation is said to be class 1 relative to K, and we could restrict to class 1 representations if desired.
Definition 4.8
 (i)
The arithmetic complexity of a Fourier transform on R, denoted \(T_{G/K}(R)\), is the minimum number of arithmetic multiplications (or additions, whichever is largest) needed to compute the Fourier transform of f on R via a straightline program for an arbitrary complexvalued function f defined on G / K.
 (ii)The reduced complexity, denoted \(t_{G/K}(R)\), is defined by$$\begin{aligned} t_G(R)=\frac{1}{\vert G/K\vert }T_{G/K}(R). \end{aligned}$$
Then the proofs of Sect. 4 extend to the following results for homogenous spaces:
Theorem 4.9
Theorem 4.10
Theorem 4.11
 (1)
\(A_1=G_1\)
 (2)
\( G_i=A_2\cdots A_i G_{i1}\) for \(2\le i\le n\).
 (3)
\(A_i\) commutes with \(G_{i2}.\)
Theorem 4.12
5 Configuration Spaces and the Maps \(*\)
In Sect. 3 we gave an overview of the SOV algorithm, assuming the existence of bilinear maps \(*\) with the properties described in part II of the SOV approach 2. In this section we determine such maps and investigate their properties.
Recall from Definition 3.3 that for a path algebra product \(x_1\cdots x_m\), \(X_i:=\mathbb {C}[\mathcal {B}_{i^+}]\cap {\text {Centralizer}}(\mathbb {C}[\mathcal {B}_{i^}]).\) We first show (Lemma 5.5 below), that each space \(X_i\) is isomorphic to the configuration space of a specific quiver \(Q_i\), with dimension \(\#{\text {Hom}}(Q_i;\mathcal {B})\).
Definition 5.1
For graded quivers Q and B, a morphism \(\phi :Q\rightarrow B\) is a mapping from arrows in Q to paths in B, along with a gradingpreserving mapping between vertices so that \(\phi (t(e))=t(\phi (e))\) and \(\phi (s(e))=s(\phi (e))\) for all arrows \(e\in E(Q)\).
Example 5.2
For Q, B as in Fig. 22, let \(\phi :Q\rightarrow B\) send the arrow \(e_1\) to the path \(f_3\circ f_2\circ f_1\).
For two graded quivers Q and B, let \({\text {Hom}}(Q;B)\) denote the set of morphisms from Q to B. For Q, R, and B graded quivers such that Q is a subquiver of R, let \({\text {Hom}}(Q\uparrow R;B)\) denote the set of morphisms from Q to B that extend to R.
Definition 5.3
The configuration space associated to Q and R relative to B, denoted \(A(Q\uparrow R;B)\), is the space of finitely supported formal \(\mathbb {C}\)linear combinations of morphisms in \({\text {Hom}}(Q\uparrow R;B)\).
Note 5.4
When \(Q=R\), we simplify notation by writing A(Q; B). If Q is a finite subquiver of R and B is locally finite, i.e. each vertex has finitely many neighbors, then \(\#{\text {Hom}}(Q\uparrow R; B)=\dim A(Q\uparrow R; B)\).
Lemma 5.5
Let \(\{\mathbb {C}[G_i]\}\) be a chain of group algebras with corresponding Bratteli diagram \(\mathcal {B}\) of highest grading at least n. Consider the quivers \(Q_{n0}\) and \(Q_{ji}^n\) of Fig. 23, along with the subquiver \(Q_{ji}\) of \(Q_{ji}^n\) consisting of the two vertices at level i and level j, along with the two paths from level i to level j:
 (i)
\(A(Q_{n0}; \mathcal {B})\cong \mathbb {C}[\mathcal {B}_n]\),
 rm (ii)
\(A(Q_{ji}; \mathcal {B})=A(Q_{ji}\uparrow Q_{ji}^n; \mathcal {B})\cong \mathbb {C}[\mathcal {B}_j]\cap {\text {Centralizer}}(\mathbb {C}[\mathcal {B}_i]).\)
Proof
This follows from an application of standard facts about Gel’fand–Tsetlin bases. For futher details, see e.g. [28, Lemma 4.1], [19, Proposition 2.3.12]. \(\square \)
By Lemma 5.5, for \(X_i=\mathbb {C}[\mathcal {B}_{i^+}]\cap {\text {Centralizer}}(\mathbb {C}[\mathcal {B}_{i^}])\) as in Definition 3.3 and for \(Q_i:=Q_{i^+i^}\), \(X_i\cong A(Q_{i}; \mathcal {B}).\)
5.1 The Bilinear Maps ‘\(*\)’
Definition 5.6
For a graded quiver R with subquivers \(Q_1, Q_2\), the symmetric difference of \(Q_1\) and \(Q_2\) is \(Q_1\triangle Q_2=(Q_1\setminus (Q_1\cap Q_2))\cup (Q_2\setminus (Q_1\cap Q_2)).\)
Example 5.7
The quiver \(Q_2\triangle Q_3\) in Fig. 9 is the symmetric difference of \(Q_2\) and \(Q_3\), while the quivers \(Q_1^\sigma \triangle Q_2^\sigma \) in Figs. 13 and 19 show the symmetric difference of \(Q_1^\sigma \) and \(Q_2^\sigma \).
Definition 5.8
Note 5.9
It is clear from the definition that the restricted product is bilinear and commutative. In “Appendix 2” we show that the restricted product is associative.
Lemma 5.10
For B a locally finite graded quiver, R a graded quiver with finite subquivers \(Q_1\) and \(Q_2\), \(f\in A(Q_1\uparrow R;B)\), and \(g\in A(Q_2\uparrow R;B)\), the restricted product \(f*g\) requires at most \(\#{\text {Hom}}((Q_1\cup Q_2)\uparrow R; B)\) scalar multiplications and at most \(\#{\text {Hom}}((Q_1\cup Q_2)\uparrow R; B)\#{\text {Hom}}((Q_1\triangle Q_2)\uparrow R; B)\) scalar additions.
Proof
To compute \(f*g\), first compute \((f\vert _{\eta \circ {\iota _1}})(g\vert _{\eta \circ {\iota _2}})\) for each \(\eta \in {\text {Hom}}(Q_1\cup Q_2\uparrow R; B)\). This requires \(\#{\text {Hom}}(Q_1\cup Q_2\uparrow R; B)\) scalar multiplications.
Next note that a scalar addition comes from each pair \(\eta _i,\eta _j\in {\text {Hom}}(Q_1\cup Q_2\uparrow R; B)\) with \(\eta _i\downarrow _{Q_1\triangle Q_2}=\eta _j\downarrow _{Q_1\triangle Q_2}=\tau \in {\text {Hom}}(Q_1\triangle Q_2\uparrow R; B);\) in total, \(\#{\text {Hom}}((Q_1\cup Q_2)\uparrow R; B)\#{\text {Hom}}((Q_1\triangle Q_2)\uparrow R; B)\) scalar additions. \(\square \)
Lemma 5.5 gives a correspondence between \(\mathbb {C}[\mathcal {B}_n]\) and the configuration space of the associated quiver \(Q_{n0}\). With Theorem 5.11 below, we see that under this isomorphism multiplication of path algebra elements corresponds to restricted products in the associated configuration spaces.
Theorem 5.11
Let \(\mathcal {B}\) be a Bratteli diagram of highest grading at least n and let \(f,g\in \mathbb {C}[\mathcal {B}_n]\). Let \(Q_1\) and \(Q_2\) be the quivers of Fig. 24 with paths p, q, and \(p'\), \(q'\), respectively. Let \(q=p'\) and let \(R=Q_1\cup Q_2\).
Proof
For P, Q paths of length n in \(\mathcal {B}\), let \(\gamma _{PQ}\in {\text {Hom}}(Q_1;\mathcal {B})\) denote the morphism that sends p to P and q to Q. Similarly, let \(\mu _{PQ}\in {\text {Hom}}(Q_2;\mathcal {B})\) (respectively, \(\tau _{PQ}\in {\text {Hom}}(Q1\triangle Q_2;\mathcal {B})\)) denote the morphism that sends \(p'\) to P and \(q'\) to Q (respectively, p to P and \(q'\) to Q).
Corollary 5.12
For e the identity element of \(\mathbb {C}[\mathcal {B}_{n}]\), a restricted product with \(\phi (e)\) requires no arithmetic operations to compute.
5.2 Use of ‘\(*\)’ in the SOV Algorithm
In this section we combine the results of Sect. 5.1 with Sect. 3 to show how the restricted product is used in the SOV algorithm.
For \(x_i\in X_i\), let \(\tilde{x}_i:=\phi _i(x_i)\in A(Q_{i}\uparrow Q_{i}^n;\mathcal {B})\). By Theorem 5.11, for \(x_1\in X_1\) and \(x_2\in X_2\), the product \(x_1x_2\) corresponds to the restricted product \(\tilde{x}_1*\tilde{x}_2:A(Q_{1};\mathcal {B})\times A(Q_{2};\mathcal {B})\rightarrow A(Q_{1}\triangle Q_{2};\mathcal {B})\), with the paths of \(Q_{1}\) and \(Q_{2}\) identified as in Figs. 25, 26, or 27.
More generally, for \(x_i\in X_i\) and the paths of \(Q_{i}\) identified as in Figs. 25, 26, 27, the product \(x_1x_2\cdots x_m\) corresponds to the restricted product \((x_1*x_2*\cdots *x_{m1})*x_m:A(Q_1\triangle \cdots \triangle Q_{m1};\mathcal {B})\times A(Q_{m};\mathcal {B})\rightarrow A(Q_1\triangle \cdots \triangle Q_m;\mathcal {B}).\)
For \(x_i\in X_i\), let \(\tilde{x}_i=\phi _i(x_i)\) as in Theorem 5.11. By Note 5.9, commutativity of the restricted product ensures that \(\tilde{x}_1*\cdots *\tilde{x}_m=\tilde{x}_{\sigma (1)}*\cdots *\tilde{x}_{\sigma (m)}\) for any \(\sigma \in S_m\). As in Sect. 3 let \(Q_i^\sigma \) denote the quiver associated to \(X_{\sigma (i)}\).
Theorem 5.13
 1.
\(x_1\cdots x_m= \phi ^{1}((((\tilde{x}_{\sigma (1)}*\tilde{x}_{\sigma (2)})*\tilde{x}_{\sigma (3)})\cdots \tilde{x}_{\sigma (m1)})*\tilde{x}_{\sigma (m)})\)
 2.
This may be computed in at most \(\sum \nolimits _{i=1}^{m1}\dim A(Q_1^\sigma \triangle \cdots \triangle Q_i^\sigma \cup Q_{i+1}^\sigma ;\mathcal {B})\) scalar multiplications, and fewer additions.
Proof
Part 1 follows from Definition 5.8 and Theorem 5.11. To prove Part 2, apply Lemma 5.10, note that the map \(\phi ^{1}\) requires no operations to compute, and also note that for any quiver R with subquiver Q, \(\dim A(Q\uparrow R;\mathcal {B})\le \dim A(Q;\mathcal {B}).\) \(\square \)
Theorem 5.14
6 Determining the Dimension of Configuration Spaces
In Sect. 5 we developed aspects of the general quiver formalism to provide the technical bedrock for the SOV algorithm (esp., the definitions of configuration space and restricted product, and basic complexity counts in terms of morphisms). The final step in computing the complexities of the algorithms outlined in Sect. 4 is to finally rewrite the morphism counts in terms of multiplicities for the restriction of representations from one group algebra to another. That is the purpose of this section. Here we accomplish this by adapting earlier work of Stanley’s on differential posets [41], a context that can also be used for Bratelli diagrams. In this section our main result is the final Corollary (Corollary 6.18) that computes \(\#{\text {Hom}}(Q,\mathcal {B})\) (for a socalled ”ntoothed quiver” Q and Bratteli diagram \(\mathcal {B}\)) in terms of spectral information from ”up” and ”down” operators on the diagram. We apply these results in Appendices 1 and 1 to give the explicit counts of Sect. 4.
6.1 General Morphism Counts
Recall from Note 5.4 that if Q is a finite subquiver of R and B a locally finite quiver, \(\dim A(Q\uparrow R; B)=\#{\text {Hom}}(Q\uparrow R; B).\) In the SOV approach, B is the Bratteli diagram associated to a chain of semisimple algebras and hence locally finite, so in this section we give results to count \(\#{\text {Hom}}(Q\uparrow R;B)\).
For B a locally finite graded quiver, \(\alpha ,\beta \in V(B)\), let \(M_B(\alpha ,\beta )\) denote the number of paths from \(\beta \) to \(\alpha \) in B. Note that for \(\mathcal {B}\) a Bratteli diagram, \(\alpha ,\beta \in V(\mathcal {B})\) correspond to irreducible representations \(\gamma \), \(\rho \) and \(M_\mathcal {B}(\alpha ,\beta )=M(\gamma , \rho ),\) as in Definition 2.12.
Theorem 6.1
Proof
A morphism specifies the image of each vertex and each arrow. This may be counted by first fixing the image of each vertex and counting all possible arrow images, then varying over all possible images of V(Q). \(\square \)
 1.
label each vertex \(\alpha _i\in Q^i\) with a vertex \(\alpha _i'\in B^i\) such that this labeling could extend to a map from R into B;
 2.
label each edge of Q from \(\beta \) to \(\alpha \) by \(M_B(\alpha ',\beta ')\);
 3.
multiply the labels and sum over all possible labellings.
Example 6.2
To further simplify counts, we first ‘smooth’ quivers before counting morphisms, i.e. we remove superfluous vertices (see Corollary 7.12 in “Smoothing Quivers” section of “Appendix 3”).
Example 6.3
6.2 Morphisms into Locally Free Bratteli Diagrams
In Sect. 6.1 we obtained general quiver morphism counting results for B a locally finite quiver. For locally free Bratteli diagrams we rewrite these results in terms of the dimensions of the corresponding subalgebras.
Definition 6.4
A Bratteli diagram \(\mathcal {B}\) is locally free if for each \(i\ge 1\), \(\mathbb {C}[\mathcal {B}_i]\) is free as a module over \(\mathbb {C}[\mathcal {B}_{i1}]\).
Example 6.5
For G a finite group, the Bratteli diagram associated to the chain of group algebras \(\mathbb {C}[G]=\mathbb {C}[G_n]>\cdots > \mathbb {C}[G_0]=\mathbb {C}\) is locally free.
Note 6.6
As the vertices of \(\mathcal {B}\) are labeled by the irreducible representations of \(\mathbb {C}[\mathcal {B}_{i}]\), elements of \(\mathbb {C}[\mathcal {B}^i]\) correspond to representations of the path algebra \(\mathbb {C}[\mathcal {B}_{i}]\). In this context, U is induction and D restriction (see [19, Proposition 2.3.12]).
Example 6.7
In Corollary 6.18 we give explicit formulas for these inner products. For \(\alpha \in V(B)\) and \(\hat{0}\) the root of \(\mathcal {B}\), let \(d_\alpha =M_\mathcal {B}(\alpha ,0)\) and let \(d_i=\sum _{\alpha \in \mathcal {B}^i} d_\alpha \alpha \).
Lemma 6.8
 (i)
For \(\alpha \in \mathcal {B}^i\), \(\langle d_i,\alpha \rangle = d_\alpha .\)
 (ii)
\(d_i=U^i\hat{0},\)
Proof
Clear from definition and induction. \(\square \)
Proposition 6.9
 (i)
\(\mathcal {B}\) is locally free.
 (ii)For each i and all \(\beta \in \mathcal {B}^{i1}\), there exists \(\lambda _i\in \mathbb {C}\) such that$$\begin{aligned} \sum _{\alpha \in \mathcal {B}^i} M_\mathcal {B}(\alpha ,\beta )d_\alpha =\lambda _i d_\beta . \end{aligned}$$
 (iii)
For each i, \(d_i\) is an eigenvector of DU.
 (iv)
For each i there exists \(\lambda _i\in \mathbb {C}\) with \(DU^i\hat{0}=\lambda _iU^{i1}\hat{0}\).
Proof
As this proof comes down to definitions and the fact that D is restriction (cf. Note 6.6), we defer it to “Properties of Locally Free Quivers” section of “Appendix 3”.
Corollary 6.10
 (i)
\(\displaystyle \lambda _i=\frac{\dim _\mathbb {C}\mathbb {C}[\mathcal {B}_i]}{\dim _\mathbb {C}\mathbb {C}[\mathcal {B}_{i1}]}\),
 (ii)
\(\displaystyle \dim _\mathbb {C}\mathbb {C}[\mathcal {B}_i]=\prod _{j=1}^i\lambda _j\).
Example 6.11
Theorem 6.13 below generalizes Theorem 3.7 of [41] and Theorem 2.3 of [42].
Definition 6.12
Let \(w=w_l\cdots w_1\) be a word in U and D and let \(\mathcal {S}=\{i\mid w_i=D\}.\) For each \(i\in \mathcal {S}\), let \(a_i=\# \{D\text {'s in} \;w\; \text {to the right of} \;w_i\},\) and similarly let \(b_i=\# \{U\text {'s in}\; w\; \text {to the right of} \;w_i\}.\) If \(b_ia_i\ge 0\) for all \(i\in \mathcal {S}\), we call w an admissible word.
Theorem 6.13
Proof
For \(\mathcal {B}\) a locally free Bratteli diagram and Q an ntoothed quiver, Theorem 6.13 allows us to determine \(\#{\text {Hom}}(Q; \mathcal {B})\).
Definition 6.14
A quiver Q is ntoothed if it consists of \(2n+1\) (not necessarily distinct) vertices \(\gamma _0,\dots ,\gamma _n,\beta _1,\dots ,\beta _n\) and distinct arrows connecting \(\gamma _{i1}\) to \(\beta _i\) and \(\gamma _i\) to \(\beta _i\).
Example 6.15
The quiver of Fig. 30 is an example of a 3toothed quiver.
Example 6.16
The quiver \(Q_1\) of Fig. 28 is 2toothed, with \(\gamma _0=\alpha _0, \gamma _1=\alpha _3, \gamma _2=\alpha _0, \beta _1=\alpha _4, \beta _2=\alpha _5\) .
Theorem 6.17
Proof
Follows from Theorem 6.1 and induction.
Corollary 6.18
Example 6.19
We use Corollary 6.18 in “Appendices 4” and “5” to give many of the complexity results needed for the proofs in Sect. 4.
7 Further Directions
The SOV approach produces savings by first treating the Fourier transform as a collection of scalar equations and then recursively structuring the summation so as to collect together irreducible matrix elements, viewed under the translation to the path algebra as pairs of paths. Through this translation, a sequence of multiplications becomes a sequence of bilinear maps indexed by subgraphs. Efficiency counts are determined by the size of the factorization sets needed for these multiplications, as well as the number of occurrences of these subgraphs in the Bratteli diagram. The resultant savings are dependent on the choice of factorization as well as combinatorial pathcounting methods used to provide the bounds in “Appendices 4” and “5”. Different choices of subgroups could provide better bounds, and in fact some applications of the Fourier transform require particular chains of parabolic subgroups [12, 14, 29], which we will investigate in further work.
In addition, our results can be generalized beyond Fourier transforms on groups. In fact, the path algebra isomorphism of Corollary 2.16 is true for the Bratteli diagram associated to any semisimple algebra. In work being prepared for publication, we extend the SOV approach to Fourier transforms on semisimple algebras and determine complexity results for the Hecke, Brauer and Birman–Wenzl–Murakami algebras [35].
Footnotes
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