Separation of Variables and the Computation of Fourier Transforms on Finite Groups, II

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.

Appendices

Appendix 1: Gel’fand–Tsetlin Bases and Adapted Representations

In Sect. 2 we introduce adapted bases and systems of Gel’fand Tsetlin bases. Here we make the formal connection between adapted bases of a group algebra chain and systems of Gel’fand Tsetlin bases for the corresponding chain of path algebras.

Definition 7.1

Given a Bratteli diagram $\mathcal {B}$, a representation of $\mathbf {\mathcal {B}}$ assigns to each $\alpha \in V(\mathcal {B})$ a linear space $V_\alpha$ and to each edge $e\in E(\mathcal {B})$ a linear map $L_e:V_{s(e)}\rightarrow V_{t(e)}$. Given two representations $(\{V_\alpha \}_{\alpha \in V(\mathcal {B})}, \{L_e\}_{e\in E(\mathcal {B})})$, $(\{W_\alpha \}_{\alpha \in V(\mathcal {B})}, \{S_e\}_{e\in E(\mathcal {B})})$, of $\mathcal {B}$, a morphism $m:V\rightarrow W$ is a family of linear maps $\{m_\alpha :V_\alpha \rightarrow W_\alpha \}_{\alpha \in V(\mathcal {B})}$ such that the diagram

$$\begin{aligned} \begin{array}{llll} &{}V_{s(e)}&{}\xrightarrow {\;L_e\;} &{}V_{t(e)}\\ m_{s(e)}&{}\downarrow &{}&{}\downarrow \;\;m_{t(e)}\\ &{}W_{s(e)}&{}\xrightarrow {\;S_e\;}&{}W_{t(e)} \end{array} \end{aligned}$$

commutes for all $e\in E(\mathcal {B})$.

A model representation of $\mathbf {\mathcal {B}}$ is a representation of $\mathcal {B}$ such that for all $e\in E(\mathcal {B})$, $L_e$ is injective, and for all nonroot vertices $\alpha \in V(\mathcal {B})$,

$$\begin{aligned} V_\alpha =\displaystyle \bigoplus _{t(e)=\alpha } Im(L_e). \end{aligned}$$

Definition 7.2

Given a chain of group algebras $\{\mathbb {C}[G_i]\}$, a model representation for $\{\mathbb {C}[G_i]\}$ is a model representation of the corresponding Bratteli diagram $\mathcal {B}$ such that:

• (i) for each $\alpha \in V(\mathcal {B})$ at level i, $V_\alpha$ is the representation space of the representation of $\mathbb {C}[G_i]$ corresponding to $\alpha$,

• (ii) for each $e\in E(\mathcal {B})$ from level i to level $i+1$, $L_e$ is $\mathbb {C}[G_i]$-equivariant, i.e., for $\rho _{s(e)}$ the representation of $\mathbb {C}[G_i]$ corresponding to s(e) and $\rho _{t(e)}$ the representation of $\mathbb {C}[G_{i+1}]$ corresponding to t(e), the diagram

  $$\begin{aligned} \begin{array}{llll} &{}V_{s(e)}&{}\xrightarrow {\;L_e\;} &{}V_{t(e)}\\ \rho _{s(e)}&{}\downarrow &{}&{}\downarrow \;\;\rho _{t(e)}\\ &{}V_{s(e)}&{}\xrightarrow {\;L_e\;}&{}V_{t(e)} \end{array} \end{aligned}$$

  commutes for all $e\in E(\mathcal {B})$.

A model representation of an algebra chain has a natural basis of paths:

Lemma 7.3

Given a model representation of a chain of subalgebras with Bratteli diagram $\mathcal {B}$, the collection of distinct paths in $\mathcal {B}$ from the root to a vertex $\alpha \in V(\mathcal {B})$ corresponds to a choice of basis for $V_\alpha$.

Proof

Consider the space $V_\beta$ corresponding to the root $\beta$, i.e., $V_\beta$ is the representation space of $\mathbb {C}[\mathcal {B}_0]=\mathbb {C}$, so $V_\beta$ is one-dimensional. Now let $\alpha$ be a vertex in $\mathcal {B}$ with $gr(\alpha )=1$. Then $V_\alpha =\displaystyle \displaystyle \bigoplus _{t(e)=\alpha } Im(L_e)\cong \displaystyle \bigoplus _{t(e)=\alpha }V_\beta$ since $L_e$ is injective. Induction gives the result. $\square$

Thus, given an irreducible representation $\rho$ of $\mathbb {C}[G_i]$ corresponding to a vertex $\alpha$ in the Bratteli diagram associated to the chain of group algebras, there is a basis for the representation space of $\rho$ indexed by the paths from the root to $\alpha$. We call such a basis a Gel’fand–Tsetlin basis, as in Definition 2.18. Given a model representation of a Bratteli diagram $\mathcal {B}$, Lemma 7.3 gives a system of Gel’fand–Tsetlin bases for $\mathcal {B}$. In fact, these are equivalent concepts:

Theorem 7.4

A system of Gel’fand–Tsetlin bases for a Bratteli diagram $\mathcal {B}$ uniquely determines a model representation for $\mathcal {B}$. Conversely, a model representation uniquely determines a system of Gel’fand–Tsetlin bases for $\mathcal {B}$.

Proof

Both require a choice of vector space for each vertex of $\mathcal {B}$, so we need only show how a choice of basis corresponds with linear maps $L_e$ for each edge e.

Given a system of bases and an edge $e\in \mathcal {B}$, a basis vector for $V_{s(e)}$ corresponds to a path P from the root to s(e). Then $e\circ P$ is a path from the root to t(e), which corresponds to a basis vector for $V_{t(e)}$. In other words, we have an injection of $V_{s(e)}$ into $V_{t(e)}$.

Conversely, given a model representation and a vertex $\alpha$, every path from the root to $\alpha$ corresponds to an injection of $\mathbb {C}$ into $V_\alpha$. Since

$$\begin{aligned} V_\alpha =\displaystyle \bigoplus _{t(e)=\alpha } Im(L_e), \end{aligned}$$

the union of the distinct images of $1\in \mathbb {C}$ over the collection of injections gives a basis for $V_\alpha$ as we vary over all possible paths from the root to $\alpha$. $\square$

Remark 7.5

The equivalent definitions of Gel’fand–Tsetlin bases and model representations coincide with the notion of a complete set of adapted representations for chains of groups. Clearly a model representation for the group algebra chain gives rise to an adapted basis since the isomorphism

$$\begin{aligned} V_\alpha =\displaystyle \bigoplus _{t(e)=\alpha } Im(L_e)\cong \displaystyle \bigoplus _{\begin{array}{c} e\in E(\mathcal {B}),\\ t(e)=\alpha \end{array}} V_{s(e)} \end{aligned}$$

describes how the representation space $V_{\alpha }$ decomposes at level $i-1$. Equivariance of the maps $L_{e}$ then gives the decomposition of the representation $\rho _\alpha$.

Further, a complete set R of inequivalent irreducible representations adapted to a chain of subgroups $G_n>G_{n-1}>\cdots >G_0$ determines the paths in the Bratteli diagram $\mathcal {B}$ of the group algebra chain by drawing $M(\rho , \gamma )$ arrows from a representation $\gamma \in R$ of $G_i$ to a representation $\rho \in R$ of $G_{i+1}$. Then a set of bases for the representation spaces of the representations in R determines a system of Gel-fand Tsetlin bases for the group algebra chain, and so by Theorem 7.4 a model representation.

Appendix 2: Restricted Product Lemmas

In this Appendix, we prove the associativity of the restricted product defined in Sect. 5.

Lemma 7.6

Let B be a locally finite graded quiver and R a graded quiver with finite subquivers $Q_1, Q_2,\dots ,Q_m$ such that $Q_i\cap Q_j\cap Q_k$ has no edges for all $i\ne j\ne k$. Let $Q_i^\triangle$ denote the quiver $Q_1\triangle \cdots \triangle Q_i$ and let $Q_i^\cup$ denote the quiver $Q_1\cup \cdots \cup Q_i$. Then for $f_i\in A(Q_i;B)$, $f_1*f_2*\cdots *f_m$ is independent of bracketing. Moreover, for $\tau \in {\text {Hom}}(Q_m^\triangle ;B)$ and $\iota _k$ the natural injection $Q_k\hookrightarrow R$,

$$\begin{aligned} (f_1*f_2*\cdots *f_m)\vert _{\tau }=\sum _{\begin{array}{c} \eta \in {\text {Hom}}(Q_m^\cup \uparrow R; B), \\ \eta \downarrow _{Q_m^\triangle }=\tau \end{array}}\prod _{k=1}^m f_k\vert _{\eta \circ \iota _k}. \end{aligned}$$

(9)

Proof

We first prove (9) inductively, as associativity clearly follows. For $n=2$, (9) is the definition of the restricted product $f_1*f_2$.

Now suppose (9) holds for $n-1$. Since $Q_i\cap Q_j\cap Q_k=\emptyset$,

$$\begin{aligned}{}[(Q_1\triangle \cdots \triangle Q_{n-1})\cup Q_n]\cap [Q_1\cup \cdots \cup Q_{n-1}]= Q_1\triangle \cdots \triangle Q_{n-1}, \end{aligned}$$

(10)

and

$$\begin{aligned}{}[(Q_1\triangle \cdots \triangle Q_{n-1})\cup Q_n]\cup [Q_1\cup \cdots \cup Q_{n-1}]= Q_1\cup \cdots \cup Q_{n}. \end{aligned}$$

(11)

By the induction hypothesis,

$$\begin{aligned} (f_1*\cdots *f_{n-1}*f_n)\vert _\tau =\sum _{\begin{array}{c} \eta \in {\text {Hom}}(Q_{n-1}^\triangle \cup Q_n\uparrow R;B)\\ \eta \downarrow _{Q_n^\triangle }=\tau \end{array}}\sum _{\begin{array}{c} \mu \in {\text {Hom}}(Q_{n-1}^\cup \uparrow R; B)\\ \mu \downarrow _{Q_{n-1}^\triangle }=\eta \downarrow _{Q_{n-1}^\triangle } \end{array}}\left( \prod _{k=1}^{n-1} f_k\vert _{\mu \circ \iota _k}\cdot f_n\vert _{\eta \circ \iota _n}\right) . \end{aligned}$$

By (10) and (11), each choice of $\mu$ and $\eta$ which agree on their intersection, the subquiver $Q_{n-1}^\triangle$, uniquely determines a morphism $\gamma \in {\text {Hom}}(Q_n^\cup \uparrow R; B)$ such that

$$\begin{aligned} \begin{array}{ll}\gamma \downarrow _{Q_1\triangle \cdots \triangle Q_{n}}=&{}\eta \downarrow _{Q_1\triangle \cdots \triangle Q_{n}}=\tau ,\\ \gamma \circ \iota _k=\mu \circ \iota _k,&{} \text {for } 1\le k\le n-1,\\ \gamma \circ \iota _n=\eta \circ \iota _n. \end{array} \end{aligned}$$

$\square$

Appendix 3: Quiver Counts

In Sect. 6., we rewrite morphism counts in terms of multiplicities of representations and dimensions of subgroup algebras (Corollary 6.18). Here we give the details needed for the proofs of Sect. 6.

1.1 Smoothing Quivers

An important simplification in morphism counts is to remove superfluous vertices from quivers, i.e., ‘smooth’ them.

Definition 7.7

A quiver B factors at level $\mathbf {i}$ if there are no arrows from a vertex $\alpha \in V(B)$ with $gr(\alpha )<i$ to a vertex $\beta \in V(B)$ with $gr(\beta )>i$.

Example 7.8

Let $\mathcal {B}$ be a Bratteli diagram with highest grading n. Then for all $0\le i \le n$, $\mathcal {B}$ factors at level i.

Definition 7.9

Let Q be a quiver with a vertex v that is the target of exactly one arrow, $e_1$, and the source of exactly one arrow, $e_2$. To smooth Q at v, remove v and replace $e_1$ and $e_2$ with an arrow from the source of $e_1$ to the target of $e_2$. To smooth Q, smooth Q at all possible v.

Example 7.10

The quiver $Q'$ of Fig. 31 results from smoothing the quiver Q.

Fig. 31

Smoothing Q

Lemma 7.11

Let B be a graded quiver that factors at level i, R a graded quiver with subquiver Q, and v a vertex of Q at level i such that Q can be smoothed at v. Let $Q'$ (respectively $R'$) be the quiver obtained by smoothing Q (respectively R) at v. Then $\#{\text {Hom}}(Q\uparrow R; B)=\#{\text {Hom}}(Q'\uparrow R'; B).$

Proof

Let $\phi \in {\text {Hom}}(Q'\uparrow R';B)$ and let f be the arrow in $Q'$ resulting from smoothing Q at v. Then f replaced two arrows, $e_1, e_2$ in Q, with $t(e_1)=v, s(e_2)=v$. Further, $s(e_1)=s(f), t(e_2)=t(f)$, so $\phi (f)$ is a path in B from a vertex $\alpha$ with $gr(\alpha )<i$ to a vertex $\beta$ with $gr(\beta )>i$. Since B factors at level i, this path contains a vertex, $v'$, with $gr(v')=i$. Let $e_1'$ be the subpath of f starting at the source of f and ending at $v'$. Similarly, let $e_2'$ be the subpath of f starting at $v'$ and ending at the target of f.

Denote by $\tilde{\phi }$ the morphism in ${\text {Hom}}(Q\uparrow R;B)$ such that:

$$\begin{aligned} \begin{array}{ll} \tilde{\phi }(e_1)=e_1',&{} \tilde{\phi }(e_2)=e_2'\\ \tilde{\phi }(e_i)=\phi (e_i),&{}\text {for } i\ne 1,2,\\ \tilde{\phi }(v_j)=\phi (v_j),&{}\text {for } v_j\ne \alpha ,\beta ,v'. \end{array} \end{aligned}$$

Clearly $\phi \rightarrow \tilde{\phi }$ is a bijection. $\square$

Corollary 7.12

Let $\mathcal {B}$ be a Bratteli diagram, R a graded quiver with subquiver Q, and $Q'$ (respectively $R'$) the quiver obtained by smoothing Q (respectively R). Then

$$\begin{aligned} \#{\text {Hom}}(Q\uparrow R; \mathcal {B})=\#{\text {Hom}}(Q'\uparrow R';\mathcal {B}). \end{aligned}$$

1.2 Properties of Locally Free Quivers

We give the details of the proofs of Proposition 6.9 and Theorem 6.13 of Sect. 6.

Proposition 7.13

(Proposition 6.9) Let $\mathcal {B}$ be a Bratteli diagram. Then the following properties are equivalent:

• (i) $\mathcal {B}$ is locally free.

• (ii) For each i and all $\beta \in \mathcal {B}^{i-1}$, 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 $DU^i\hat{0}=\lambda _iU^{i-1}\hat{0}$.

Proof

Statements (ii), (iii), and (iv) are equivalent by definition and Lemma 6.8. For example:

(iii)$\Rightarrow$ (iv) $DU^i\hat{0}=DUd_{i-1}=\lambda _id_{i-1}=\lambda _iU^{i-1}\hat{0}$

(iv)$\Rightarrow$ (iii) $DUd_i=DU^{i+1}\hat{0}=\lambda _{i+1}U^i\hat{0}=\lambda _{i+1}d_i$ We leave the remaining equivalences of (ii), (iii), and (iv) to the reader.

To show the equivalence of (i) and (iv), recall from Note 6.6 that elements of $\mathbb {C}[\mathcal {B}^i]$ correspond to representations of $\mathbb {C}[\mathcal {B}_i]$, i.e. $\mathbb {C}[\mathcal {B}_i]$-modules. Under this identification, the regular representation of $\mathbb {C}[\mathcal {B}_i]$ corresponds to the sum

$$\begin{aligned} \sum _{\alpha \in \mathcal {B}^i} d_\alpha \alpha =d_i\in \mathbb {C}[\mathcal {B}^i]. \end{aligned}$$

Since D is restriction ([19, Proposition 2.3.12]), the restriction of the regular representation of $\mathbb {C}[\mathcal {B}_i]$ to $\mathbb {C}[\mathcal {B}_{i-1}]$ corresponds to $Dd_i=DUd_{i-1}$.

(i)$\Rightarrow$(iv) For $\mathcal {B}$ locally free, $\mathbb {C}[\mathcal {B}_i]$ is free as a module over $\mathbb {C}[\mathcal {B}_{i-1}]$ with rank $\lambda _i\in \mathbb {C}$. Then the regular representation of $\mathbb {C}[\mathcal {B}_i]$ decomposes in $\mathbb {C}[\mathcal {B}_{i-1}]$ as $\lambda _i$ copies of the regular representation of $\mathbb {C}[\mathcal {B}_{i-1}]$. Thus,

$$\begin{aligned} DU^i\hat{0}=DUd_{i-1}=Dd_i=\lambda _id_{i-1}=\lambda _iU^{i-1}\hat{0}. \end{aligned}$$

(iv)$\Rightarrow$(i) $DUd_{i-1}=Dd_i=\lambda _id_{i-1}$ and so $\dim _\mathbb {C}\mathbb {C}[\mathcal {B}_i]=\lambda _i\dim _\mathbb {C}\mathbb {C}[\mathcal {B}_{i-1}]$; hence $\lambda _i$ is rational and positive. To show $\lambda _i$ integral, let

$$\begin{aligned} \begin{array}{lll} \lambda _i=\frac{p}{q},&\gcd (p,q)=1,&p,q>0. \end{array} \end{aligned}$$

Let $m=\gcd (d_\beta )$ over all $\beta \in \mathcal {B}^{i-1}$, so $\frac{d_\beta }{m}$ an integer for all $\beta \in \mathcal {B}^{i-1}$. Then

$$\begin{aligned} \begin{array}{ll} \displaystyle \frac{1}{m}\frac{p}{q}\sum _{\beta \in \mathcal {B}^{i-1}} d_\beta \beta &{}=\displaystyle \frac{1}{m}\lambda _id_{i-1}\\ &{}=\displaystyle \frac{1}{m}DUd_{i-1}\\ &{}=\displaystyle \frac{1}{m}\sum _{\alpha \in \mathcal {B}^i}\sum _{\beta \in \mathcal {B}^{i-1}}M_\mathcal {B}(\alpha ,\beta )d_\alpha \beta \\ &{}=\displaystyle \sum _{\alpha \in \mathcal {B}^i}\sum _{\beta \in \mathcal {B}^{i-1}}M_\mathcal {B}(\alpha ,\beta )^2\frac{d_\beta }{m}\beta . \end{array} \end{aligned}$$

Then the coefficient of $\beta$ is an integer and thus $q\vert \frac{d_\beta }{m}$ for all $\beta \in \mathcal {B}^i$. But $m=\gcd (d_\beta )$, and thus $q=1$, making $\lambda _i$ an integer. $\square$

Theorem 7.14

(cf. Theorem 6.13) Let $\mathcal {B}$ be a locally free Bratteli diagram and $w=D^{d_n}U^{u_n}\cdots D^{d_1}U^{u_1}$ an admissible word in U and D. Then for $s=\sum _{i=1}^n u_i-d_i$ and $\alpha \in \mathcal {B}^{s}$,

$$\begin{aligned} \langle w\hat{0},\alpha \rangle =d_\alpha \prod _{i\in \mathcal {S}}\lambda _{b_i-a_i}. \end{aligned}$$

Proof

To be admissible, $d_{i},u_{i}> 0$ for all $1\le i\le n$ and

$$\begin{aligned} \sum _{j=1}^i d_j\le \sum _{j=1}^i u_j. \end{aligned}$$

Let $\mathcal {S}_k=\{i\in \mathcal {S}\vert i\le \sum _{j=1}^k (d_j+u_j)\}$ let $s_k=\sum _{i=1}^k u_i-d_i$, and let $w_k=D^{d_k}U^{u_k}\cdots D^{d_1}U^{u_1}$. We prove inductively that

$$\begin{aligned} w_k\hat{0}=\prod _{i\in \mathcal {S}_k}\lambda _{b_i-a_i}\sum _{\alpha \in \mathcal {B}^{s_k}} d_\alpha \alpha . \end{aligned}$$

Note that $w_1=D^{d_1}U^{u_1}$. Then $\mathcal {S}_1=\{u_1+1,u_1+2\dots ,u_1+d_1\}$ and for all $i\in \mathcal {S}_1$, $b_i=u_1$ and $a_i=i-u_1-1$. By Proposition 6.9, Lemma 6.8, and induction,

$$\begin{aligned} w_1\hat{0}=\displaystyle \prod _{i\in \mathcal {S}_1}\lambda _{b_i-a_i}\sum _{\alpha \in \mathcal {B}^{s_1}}d_\alpha \alpha \end{aligned}$$

Now suppose true for $n-1$. Then

$$\begin{aligned} \begin{array}{ll} w\hat{0}&{}=w_n\hat{0}\\ &{}=D^{d_n}U^{u_n}w_{n-1}\hat{0}\\ &{}=D^{d_n}U^{u_n}\displaystyle \prod _{i\in \mathcal {S}_{n-1}}\lambda _{b_i-a_i}U^{s_{n-1}}\hat{0}\\ &{}=\displaystyle \prod _{i\in \mathcal {S}_{n-1}}\lambda _{b_i-a_i}D^{d_n}U^{u_n+s_{n-1}}\hat{0}, \end{array} \end{aligned}$$

and the same argument as in the base case gives the result. $\square$

Appendix 4: Combinatorial Lemmas for the Weyl Groups

The SOV approach reduces Theorem 1.1 (respectively, Theorem 1.2) to counting the number of morphisms of the quivers of Fig. 14 into the Bratteli diagram of $B_n$ (respectively, $D_n$). In this section we consider the Bratteli diagrams associated to $B_n$ and $D_n$ to provide the bounds used in the proofs of Theorems 1.1 and 1.2. Note that Lemmas 7.16, 7.17, 7.20, 7.21 and Corollaries 7.19 and 7.22 all hold for $n\ge 2$, $i\ge 2$.

1.1 The Weyl Group $B_n$

The Bratteli diagram $\mathcal {B}$ associated to the chain $\mathbb {C}[B_n]>\mathbb {C}[B_{n-1}]>\cdots > \mathbb {C}$ is a generalization of Young’s diagram — inequivalent irreducible representations of $\mathbb {C}[B_i]$ are indexed by pairs of partitions $(\lambda _1,\lambda _2)$, of k and l, respectively, with $k+l=i$. Pairs $(\lambda _1,\lambda _2)$, $(\mu _1,\mu _2)$ are connected by an edge if either $\lambda _1$ may be obtained from $\mu _1$ by adding a box, or if $\lambda _2$ may be obtained from $\mu _2$ by adding a box [36] (see Fig. 32). Note that this is a multiplicity-free diagram.

Fig. 32

Bratteli diagram for $B_2$

Theorem 7.15

For $\mathcal {J}_i^n$ as in Fig. 14 and $\mathcal {B}$ the Bratteli diagram associated to the Weyl group $B_n$, $\# {\text {Hom}}(\mathcal {J}_i^n\uparrow \mathcal {Q};\mathcal {B})\le 2\vert B_n\vert$.

Proof

By Theorem 6.1, we see that $\# {\text {Hom}}(\mathcal {J}_i^n\uparrow \mathcal {Q};\mathcal {B})$ is equal to the sum

$$\begin{aligned} \begin{array}{l} \displaystyle \sum _{\alpha _j,\beta _j\in \mathcal {B}^j}M_\mathcal {B}(\beta _{n},\beta _{i})M_\mathcal {B}(\beta _{i},\beta _{i-1})M_\mathcal {B}(\beta _{i-1},\alpha _{i-2})M_\mathcal {B}(\beta _{i},\alpha _{i-1})M_\mathcal {B}(\alpha _{i-1},\alpha _{i-2})d_{\alpha _{i-2}}d_{\beta _{n}}\\ \displaystyle \quad =\sum _{\alpha _j,\beta _j\in \mathcal {B}^j}M_\mathcal {B}(\beta _{n},\beta _{i})M_\mathcal {B}(\beta _{i},\alpha _{i-2})M_\mathcal {B}(\beta _{i},\alpha _{i-1})M_\mathcal {B}(\alpha _{i-1},\alpha _{i-2})d_{\alpha _{i-2}}d_{\beta _{n}}\\ \displaystyle \quad \le M_\mathcal {B}(B_i,B_{i-2})\sum _{\alpha _j,\beta _j\in \mathcal {B}^j}M_\mathcal {B}(\beta _{n},\beta _{i}) M_\mathcal {B}(\beta _{i},\alpha _{i-1})M_\mathcal {B}(\alpha _{i-1},\alpha _{i-2})d_{\alpha _{i-2}}d_{\beta _n}, \end{array} \end{aligned}$$

for $M_\mathcal {B}(B_i,B_j):=\max M_\mathcal {B}(\alpha _i,\alpha _j)$ over all $\alpha _i\in \mathcal {B}^i, \alpha _j\in \mathcal {B}^j$. By Corollary 6.18,

$$\begin{aligned} \begin{array}{ll} \# {\text {Hom}}(\mathcal {J}_i^n\uparrow \mathcal {Q};\mathcal {B})&{}= M_{\mathcal {B}}(B_{i},B_{i-2}) \langle D^{n}U^{n}\hat{0},\hat{0}\rangle \\ &{}=M_{\mathcal {B}}(B_{i},B_{i-2})\vert B_{n}\vert .\end{array} \end{aligned}$$

Lemma 7.16 below shows that $M_{\mathcal {B}}(B_{i},B_{i-2})\le 2.$ Thus

$$\begin{aligned} \#{\text {Hom}}(\mathcal {J}_i^n\uparrow \mathcal {Q};\mathcal {B})\le 2\vert B_n\vert . \end{aligned}$$

$\square$

Lemma 7.16

$M_\mathcal {B}(B_i,B_{i-2})\le 2$

Proof

Suppose not. Then since $\mathcal {B}$ is multiplicity-free, we must have distinct pairs of partitions $\mathbf {\kappa }=(\kappa _1,\kappa _2),$, $\mathbf {\rho }=(\rho _1,\rho _2),$, $\mathbf {\gamma }=(\gamma _1,\gamma _2),$ $\mathbf {\eta }=(\eta _1,\eta _2),$ and $\mathbf {\lambda }=(\lambda _1,\lambda _2)$ as in Fig. 33.

Fig. 33

Subquiver of $\mathcal {B}$ if $M_\mathcal {B}(B_i,B_{i-2})> 2$

Pairs of partitions are adjacent in $\mathcal {B}$ if one is acquired from the other by adding a single box; hence either $\rho _1=\lambda _1$ or $\rho _2=\lambda _2$. The same holds for $\eta ,\gamma$. Similarly, $\kappa _1=\rho _1$ or $\kappa _2=\rho _2$ and the same holds for $\eta , \gamma$.

Without loss of generality, we need only consider the following two cases:

Case 1: $\rho _1,\eta _1,\gamma _1=\lambda _1$.

If $\kappa _1\ne \lambda _1$, then $\kappa _2=\rho _2,\gamma _2,\eta _2$, but then $\mathbf {\rho }=\mathbf {\eta }=\mathbf {\gamma }$, a contradiction.

If $\kappa _1=\lambda _1$ then $\kappa _2$ is obtained from $\lambda _2$ by adding two boxes, which may be done in at most two ways, so $\eta ,\gamma ,\rho$ are not all distinct, a contradiction.

Case 2: $\rho _1=\lambda _1=\eta _1$ and $\gamma _2=\lambda _2$.

If $\kappa _1=\lambda _1$ then since $\gamma _1\ne \lambda _1$, we see that $\kappa _1\ne \gamma _1$. Thus, $\kappa _2=\gamma _2$. But then $(\kappa _1,\kappa _2)=(\lambda _1,\lambda _2)$, a contradiction.

Now if $\kappa _1\ne \lambda _1$, then $\kappa _2=\rho _2,\eta _2$, but then $\rho =\eta$, a contradiction. $\square$

The following two lemmas provide a bound for $\dim A(\mathcal {H}_i^n\uparrow G;\mathcal {B})$, for $\mathcal {H}_i^n$ as in Fig. 14.

Lemma 7.17

• (1) $\displaystyle \#{\text {Hom}}(\mathcal {H}_i^n\uparrow G;\mathcal {B})=\frac{\vert B_{n-1}\vert }{\vert B_{i-1}\vert }\#{\text {Hom}}(\mathcal {H}_i^i\uparrow G;\mathcal {B})$

• (2)
  $$\begin{aligned} \#{\text {Hom}}(\mathcal {H}_i^i\uparrow G;\mathcal {B})= & {} \quad 2(i-1)\vert B_{i-1}\vert + \sum _{\begin{array}{c} (\beta _{i-1}^1,\beta _{i-1}^2)=\\ \mathbf {\beta }_{i-1}\in \mathcal {B}^{i-1} \end{array}} ({\text {jmp}}(\beta _{i-1}^1)\\&+{\text {jmp}}(\beta _{i-1}^2))({\text {jmp}}(\beta _{i-1}^1)+{\text {jmp}}(\beta _{i-1}^2)+1)d_{\beta _{i-1}}^2, \end{aligned}$$

where ${\text {jmp}}$ denotes the jump of a partition, i.e, the number of ways to remove a single box to form a new partition.

Proof

To prove (1), first note by Theorem 6.1, $\# {\text {Hom}}(\mathcal {H}_i^n\uparrow G;\mathcal {B})$ equals

$$\begin{aligned} \begin{array}{lll} \displaystyle \sum _{\alpha _j,\beta _j\in \mathcal {B}^j}M_\mathcal {B}(\beta _{n-1},\beta _{i-1})M_\mathcal {B}(\beta _{i-1},\beta _{i-2})M_\mathcal {B}(\alpha _{i},\alpha _{i-1})M_\mathcal {B}(\alpha _{i},\beta _{i-1})\\ \quad \times M_\mathcal {B}(\alpha _{i-1},\beta _{i-2})d_{\alpha _{i-1}}d_{\beta _{n-1}}. \end{array} \end{aligned}$$

By Corollary 6.18,

$$\begin{aligned} \begin{array}{ll} \displaystyle \sum _{\beta _{n-1}\in \mathcal {B}^{n-1}} M_{\mathcal {B}}(\beta _{n-1},\beta _{i-1})d_{\beta _{n-1}}&{}= \langle D^{n-i}U^{n-1}\hat{0},\beta _{i-1}\rangle \\ &{}= \lambda _{n-1}\lambda _{n-2}\cdots \lambda _id_{\beta _{i-1}}\\ &{}=\displaystyle \frac{\vert B_{n-1}\vert }{\vert B_{i-1}\vert }d_{\beta _{i-1}}. \end{array} \end{aligned}$$

Then $\# {\text {Hom}}(\mathcal {H}_i^n\uparrow G;\mathcal {B})$ equals

$$\begin{aligned}&\displaystyle \frac{\vert B_{n-1}\vert }{\vert B_{i-1}\vert }\sum _{\alpha _j,\beta _j\in \mathcal {B}^j}M_\mathcal {B}(\beta _{i-1},\beta _{i-2})M_\mathcal {B}(\alpha _{i},\alpha _{i-1})M_\mathcal {B}(\alpha _{i},\beta _{i-1}) M_\mathcal {B}(\alpha _{i-1},\beta _{i-2})d_{\beta _{i-1}}d_{\alpha _{i-1}}\\&\quad =\displaystyle \frac{\vert B_{n-1}\vert }{\vert B_{i-1}\vert }\#{\text {Hom}}(\mathcal {H}_i^i\uparrow G;\mathcal {B}). \end{aligned}$$

To prove (2),

$$\begin{aligned} \begin{array}{ll} \#{\text {Hom}}(\mathcal {H}_i^i\uparrow G;\mathcal {B})&{}=\displaystyle \sum _{\alpha _j,\beta _j\in \mathcal {B}^j}M_\mathcal {B}(\beta _{i-1},\beta _{i-2})M_\mathcal {B}(\alpha _{i},\alpha _{i-1}) \\ &{}\quad M_\mathcal {B}(\alpha _{i},\beta _{i-1})M_\mathcal {B}(\alpha _{i-1},\beta _{i-2})d_{\beta _{i-1}}d_{\alpha _{i-1}}\\ &{}=\displaystyle \sum _{\alpha _{i-1}\ne \beta _{i-1}}+\sum _{\alpha _{i-1}=\beta _{i-1}}, \end{array} \end{aligned}$$

for $\displaystyle \sum _{\alpha _{i-1}\ne \beta _{i-1}}=\sum _{\begin{array}{c} \alpha _j,\beta _j\in \mathcal {B}^j\\ \alpha _{i-1}\ne \beta _{i-1} \end{array}}M_\mathcal {B}(\beta _{i-1},\beta _{i-2})M_\mathcal {B}(\alpha _{i},\alpha _{i-1}) M_\mathcal {B}(\alpha _{i},\beta _{i-1})M_\mathcal {B}(\alpha _{i-1},\beta _{i-2})d_{\beta _{i-1}}d_{\alpha _{i-1}}$ and

$\displaystyle \sum _{\alpha _{i-1}=\beta _{i-1}}=\sum _{\begin{array}{c} \alpha _j,\beta _j\in \mathcal {B}^j\\ \alpha _{i-1}=\beta _{i-1} \end{array}}M_\mathcal {B}(\beta _{i-1},\beta _{i-2})^2M_\mathcal {B}(\alpha _{i},\beta _{i-1})^2(d_{\beta _{i-1}})^2.$

First suppose $\mathbf {\alpha }_{i-1}=(\alpha _{i-1}^1,\alpha _{i-1}^2),\mathbf {\beta }_{i-1}=(\beta _{i-1}^1,\beta _{i-1}^2)$ are distinct pairs of partitions. Then they jointly determine $\mathbf {\alpha }_i=(\alpha _i^1,\alpha _i^2)$. Thus, the sum $\sum \nolimits _{\alpha _{i-1}\ne \beta _{i-1}}$ becomes

$$\begin{aligned} \begin{array}{ll} &{}\displaystyle \sum _{\begin{array}{c} \beta _{i-2}\in \mathcal {B}^{i-2}\\ \alpha _{i-1}\ne \beta _{i-1}\in \mathcal {B}^{i-1} \end{array}}M_\mathcal {B}(\beta _{i-1},\beta _{i-2})M_\mathcal {B}(\alpha _{i-1},\beta _{i-2})d_{\alpha _{i-1}}d_{\beta _{i-1}}\\ =&{}\displaystyle \sum _{\begin{array}{c} \beta _{i-2}\in \mathcal {B}^{i-2}\\ \alpha _{i-1},\beta _{i-1}\in \mathcal {B}^{i-1} \end{array}} M_\mathcal {B}(\beta _{i-1},\beta _{i-2})M_\mathcal {B}(\alpha _{i-1},\beta _{i-2})d_{\beta _{i-1}}d_{\alpha _{i-1}} \\ &{}-\displaystyle \sum _{\begin{array}{c} \beta _{j}\in \mathcal {B}^{}\\ \alpha _{i-1}=\beta _{i-1} \end{array}}M_\mathcal {B}(\beta _{i-1},\beta _{i-2})^2(d_{\beta _{i-1}})^2\\ =&{}\displaystyle \frac{\vert B_{i-1}\vert }{\vert B_{i-2}\vert }\sum _{\beta _{j},\alpha _j\in \mathcal {B}^{j}}{M_\mathcal {B}(\alpha _{i-1},\beta _{i-2})d_{\beta _{i-2}}d_{\alpha _{i-1}}}-\sum _{\begin{array}{c} \beta _{j}\in \mathcal {B}^{j}\\ \alpha _{i-1}=\beta _{i-1} \end{array}}M_\mathcal {B}(\beta _{i-1},\beta _{i-2})^2(d_{\beta _{i-1}})^2\\ =&{}\displaystyle \frac{\vert B_{i-1}\vert }{\vert B_{i-2}\vert }\sum _{\alpha _{i-1}\in \mathcal {B}^{i-1}}(d_{\alpha _{i-1}})^2-\sum _{\begin{array}{c} \beta _{j}\in \mathcal {B}^{j}\\ \alpha _{i-1}=\beta _{i-1} \end{array}}M_\mathcal {B}(\beta _{i-1},\beta _{i-2})^2(d_{\beta _{i-1}})^2\\ =&{}\displaystyle \frac{{\vert B_{i-1}\vert }^2}{\vert B_{i-2}\vert }-\sum _{\beta _{i-1}\in \mathcal {B}^{i-1}}({\text {jmp}}(\beta _{i-1}^1)+{\text {jmp}}(\beta _{i-1}^2))(d_{\beta _{i-1}})^2, \end{array} \end{aligned}$$

and so

$$\begin{aligned} \begin{array}{l} \displaystyle \sum _{\alpha _{i-1}\ne \beta _{i-1}} =\displaystyle 2(i-1)\vert B_{i-1}\vert -\sum _{\beta _{i-1}\in \mathcal {B}^{i-1}}({\text {jmp}}(\beta _{i-1}^1)+{\text {jmp}}(\beta _{i-1}^2))(d_{\beta _{i-1}})^2. \end{array} \end{aligned}$$

(12)

Now suppose $\alpha _{i-1}=\beta _{i-1}$. Then $\alpha _{i}$ is obtained from $\beta _{i-1}$ by adding a box to $\beta _{i-1}^1$ or $\beta _{i-1}^2$, while $\beta _{i-2}$ is obtained from $\beta _{i-1}$ by removing a box from $\beta _{i-1}^1$ or $\beta _{i-1}^2$. Thus,

$$\begin{aligned} \sum _{\alpha _{i-1}=\beta _{i-1}}=\sum _{\beta _{i-1}\in \mathcal {B}^{i-1}} ({\text {jmp}}(\beta _{i-1}^1)+{\text {jmp}}(\beta _{i-2}^1)) ({\text {jmp}}(\beta _{i-1}^1)+{\text {jmp}}(\beta _{i-2}^1)+2)(d_{\beta _{i-1}})^2.\nonumber \\ \end{aligned}$$

(13)

Summing Eqs. (12) and (13) gives (2). $\square$

Lemma 7.18

For any pair of partitions $(\beta _i^1,\beta _i^2)$ with $\vert \beta _i^1\vert +\vert \beta _i^2\vert =i$,

$$\begin{aligned} ({\text {jmp}}(\beta _{i}^1)+{\text {jmp}}(\beta _{i}^2))({\text {jmp}}(\beta _{i}^1)+{\text {jmp}}(\beta _{i}^2)+1)\le 6i \end{aligned}$$

Proof

Let $k=\vert \beta _i^1\vert$, $l=\vert \beta _i^2\vert$, $a_k={\text {jmp}}(\beta _i^1)$, and $a_l={\text {jmp}}(\beta _i^2)$. Then $k+l=i$ and by [28, Lemma 5.3],

$$\begin{aligned} \begin{array}{ll} a_k(a_k+1)\le 2k,&a_l(a_l+1)\le 2l. \end{array} \end{aligned}$$

Then

$$\begin{aligned} \begin{array}{ll} ({\text {jmp}}(\beta _{i}^1)+{\text {jmp}}(\beta _{i}^2))({\text {jmp}}(\beta _{i}^1)+{\text {jmp}}(\beta _{i}^2)+1)&{}= (a_k+a_l)(a_k+a_l+1)\\ &{}=a_k(a_k+1)+a_l(a_l+1)+2a_ka_l\\ &{}\le 2k+2l+2(2i)\\ &{}\le 6i. \end{array} \end{aligned}$$

$\square$

Combining Lemmas 7.17 and 7.18 gives the following bound:

Corollary 7.19

$\#{\text {Hom}}(\mathcal {H}_i^n\uparrow G;\mathcal {B})\le \frac{4(i-1)}{n}\vert B_n\vert .$

1.2 The Weyl Group $D_n$

The Bratteli diagram $\mathcal {B}$ associated to the chain $\mathbb {C}[D_n]> \mathbb {C}[D_{n-1}]>\cdots > \mathbb {C}$ is similar to the Bratteli diagram associated to the Weyl group $B_n$ in that irreducible representations of $\mathbb {C}[D_i]$ are indexed by pairs of partitions, $(\lambda _1,\lambda _2)$ of k and l, respectively, with $k+l=i$. However, if $\lambda _1\ne \lambda _2$, the irreducible representation indexed by $(\lambda _1,\lambda _2)$ is the same as that indexed by $(\lambda _2,\lambda _1)$. If $\lambda _1=\lambda _2=\lambda$ then two distinct irreducible representations are indexed by the pair $(\lambda ,\lambda )$, and denoted by $(\lambda ,\lambda )^+$ and $(\lambda ,\lambda )^-$ [36] (see Fig. 34). Note that this is a multiplicity-free diagram.

Fig. 34

Bratteli diagram for $D_3$

Lemma 7.20

$M_\mathcal {B}(D_i,D_{i-2})\le 3$.

Proof

Suppose not. Then since $\mathcal {B}$ is multiplicity-free, there exist pairs of partitions $\mathbf {\kappa }=(\kappa _1,\kappa _2),$ $\mathbf {\rho }=(\rho _1,\rho _2),$ $\mathbf {\gamma }=(\gamma _1,\gamma _2),$ $\mathbf {\eta }=(\eta _1,\eta _2),$ $\mathbf {\mu }=(\mu _1,\mu _2),$ and $\mathbf {\lambda }=(\lambda _1,\lambda _2)$ connected in $\mathcal {B}$ as in Fig. 35.

Fig. 35

Subquiver of $\mathcal {B}$ if $M_\mathcal {B}(D_i,D_{i-2})> 3$

However, the proof of Lemma 7.16 dictates that no three of $\eta ,\mu ,\gamma ,\rho$ are distinct pairs of partitions. Thus, without loss of generality,

$$\begin{aligned} \begin{array}{llll} (\eta _1,\eta _2)=(\alpha ,\alpha )^+,&(\mu _1,\mu _2)=(\alpha ,\alpha )^-,&(\gamma _1,\gamma _2)=(\beta ,\beta )^+,&(\rho _1,\rho _2)=(\beta ,\beta )^-, \end{array} \end{aligned}$$

for $\alpha , \beta$ distinct partitions of $\frac{j-1}{2}$. Then as in the proof of Lemma 7.16, either $\lambda _1=\eta _1=\alpha$ or $\lambda _2=\eta _2=\alpha$. Without loss, suppose $\lambda _1=\alpha$. Then since $\alpha \ne \beta$, $\lambda _2$ must be $\beta$. However, $\vert \lambda _1\vert +\vert \lambda _2\vert =\vert \alpha \vert +\vert \beta \vert >j-2$, a contradiction. $\square$

Lemma 7.20 is used in the proof of Theorem 1.2 to give a bound on $\dim A(\mathcal {J}_i^n\uparrow G;\mathcal {B})$, for $\mathcal {J}_i^n$ as in Fig. 14. The following two lemmas provide a bound for $\dim A(\mathcal {H}_i^n\uparrow G;\mathcal {B})$, for $\mathcal {H}_i^n$ as in Fig. 14.

Lemma 7.21

• (1) $\displaystyle \#{\text {Hom}}(\mathcal {H}_i^n\uparrow G;\mathcal {B})=\frac{\vert D_{n-1}\vert }{\vert D_{i-1}\vert }\#{\text {Hom}}(\mathcal {H}_i^i\uparrow G;\mathcal {B}),$

• (2) for i odd, $\#{\text {Hom}}(\mathcal {H}_i^i\uparrow G;\mathcal {B})$ is at most

  $$\begin{aligned} \begin{array}{ll} \displaystyle \frac{\vert D_{i-1}\vert ^2}{\vert D_{i-2}\vert }+\sum _{(\alpha ,\alpha )^\pm \in \mathcal {B}^{i-1}} ({\text {jmp}}(\alpha ))({\text {jmp}}(\alpha )+1)(d_{(\alpha ,\alpha )^+})^2\\ \displaystyle +\sum _{\begin{array}{c} (\beta _{i-1}^1,\beta _{i-1}^2)=\\ \mathbf {\beta }_{i-1}\in \mathcal {B}^{i-1} \end{array}} ({\text {jmp}}(\beta _{i-1}^1)+{\text {jmp}}(\beta _{i-1}^2))({\text {jmp}}(\beta _{i-1}^1)+{\text {jmp}}(\beta _{i-1}^2)+1)d_{\beta _{i-1}}^2, \end{array} \end{aligned}$$

• (3) for i even, $\#{\text {Hom}}(\mathcal {H}_i^i\uparrow G;\mathcal {B})$ is at most

$$\begin{aligned} \begin{array}{l} \displaystyle 2\displaystyle \frac{\vert D_{i-1}\vert ^2}{\vert D_{i-2}\vert }+2\displaystyle \sum _{\begin{array}{c} (\beta _{i-1}^1,\beta _{i-1}^2)=\\ \mathbf {\beta }_{i-1}\in \mathcal {B}^{i-1} \end{array}} ({\text {jmp}}(\beta _{i-1}^1)\\ \quad +{\text {jmp}}(\beta _{i-1}^2))(2{\text {jmp}}(\beta _{i-1}^1)+2{\text {jmp}}(\beta _{i-1}^2)+3)d_{\beta _{i-1}}^2, \end{array} \end{aligned}$$

where ${\text {jmp}}$ denotes the jump of a partition, i.e., the number of ways to remove a single box to form a new partition.

Proof

Part (1) follows from the proof of Lemma 7.17.

To prove (2), consider note that $\#{\text {Hom}}(\mathcal {H}_i^i\uparrow G;\mathcal {B})$ equals

$$\begin{aligned} \begin{array}{l} =\displaystyle \sum _{\alpha _j,\beta _j\in \mathcal {B}^j}M_\mathcal {B}(\beta _{i-1},\beta _{i-2})M_\mathcal {B}(\alpha _{i},\alpha _{i-1})M_\mathcal {B}(\alpha _{i},\beta _{i-1})M_\mathcal {B}(\alpha _{i-1},\beta _{i-2})d_{\beta _{i-1}}d_{\alpha _{i-1}}\\ =\displaystyle \sum _{\alpha _{i-1}\ne \beta _{i-1}}+\sum _{\begin{array}{c} \alpha _{i-1}\ne \beta _{i-1}\\ \alpha _{i-1}=(\alpha ,\alpha )^\pm =\beta _{i-1} \end{array}}+\sum _{\alpha _{i-1}=\beta _{i-1}}, \end{array} \end{aligned}$$

For

$$\begin{aligned} \begin{array}{l} \displaystyle \sum _{\alpha _{i-1}\ne \beta _{i-1}}:=\displaystyle \sum _{\begin{array}{c} \alpha _j,\beta _j\in \mathcal {B}^j\\ \alpha _{i-1}\ne \beta _{i-1} \end{array}}M_\mathcal {B}(\beta _{i-1},\beta _{i-2}) \\ M_\mathcal {B}(\alpha _{i},\alpha _{i-1})M_\mathcal {B}(\alpha _{i},\beta _{i-1})M_\mathcal {B}(\alpha _{i-1},\beta _{i-2})d_{\beta _{i-1}}d_{\alpha _{i-1}}, \end{array} \end{aligned}$$

over partitions $\alpha _{i-1}\ne \beta _{i-1}$ such that if $\alpha _{i-1}=(\alpha ,\alpha )^\pm$ then $\beta _{i-1}\ne (\alpha ,\alpha )^\pm$,

$$\begin{aligned} \begin{array}{l} \displaystyle \sum _{\begin{array}{c} \alpha _{i-1}\ne \beta _{i-1}\\ \alpha _{i-1}:=(\alpha ,\alpha )^\pm \\ =\beta _{i-1} \end{array}}:=\displaystyle \sum _{\begin{array}{c} \alpha _j,\beta _j\in \mathcal {B}^j\\ \alpha _{i-1}\ne \beta _{i-1}\\ \alpha _{i-1}=(\alpha ,\alpha )^\pm =\beta _{i-1} \end{array}}M_\mathcal {B}(\beta _{i-1},\beta _{i-2})M_\mathcal {B}(\alpha _{i},\alpha _{i-1}) \\ \qquad \times \,M_\mathcal {B}(\alpha _{i},\beta _{i-1})M_\mathcal {B}(\alpha _{i-1},\beta _{i-2})d_{\beta _{i-1}}d_{\alpha _{i-1}}, \end{array} \end{aligned}$$

and

$$\begin{aligned} \begin{array}{l} \displaystyle \sum _{\alpha _{i-1}:=\beta _{i-1}}:=\displaystyle \sum _{\begin{array}{c} \alpha _j,\beta _j\in \mathcal {B}^j\\ \alpha _{i-1}=\beta _{i-1} \end{array}}M_\mathcal {B}(\beta _{i-1},\beta _{i-2})^2M_\mathcal {B}(\alpha _{i},\beta _{i-1})^2(d_{\beta _{i-1}})^2. \end{array} \end{aligned}$$

As in the proof of Lemma 7.17,

$$\begin{aligned} \sum _{\alpha _{i-1}\ne \beta _{i-1}}\le \frac{\vert D_{i-1}\vert ^2}{\vert D_{i-2}\vert }-\sum _{\beta _{i-1}\in \mathcal {B}^{i-1}}({\text {jmp}}(\beta _{i-1}^1)+{\text {jmp}}(\beta _{i-1}^2))(d_{\beta _{i-1}})^2, \end{aligned}$$

(14)

the inequality appearing because if $\beta _{i-1}=(\alpha , \alpha )$, ${\text {jmp}}(\alpha )+{\text {jmp}}(\alpha )$ is an overestimate since $(\alpha ,\beta )$ represents the same representation as $(\beta , \alpha )$ in $\mathcal {B}$. Similarly, the proof of Lemma 7.17 gives

$$\begin{aligned} \sum _{\alpha _{i-1}=\beta _{i-1}}\!\le \!\!\sum _{\beta _{i-1}\in \mathcal {B}^{i-1}} ({\text {jmp}}(\beta _{i-1}^1)+{\text {jmp}}(\beta _{i-1}^2)) ({\text {jmp}}(\beta _{i-1}^1)+{\text {jmp}}(\beta _{i-1}^2)+2)(d_{\beta _{i-1}})^2.\!\!\!\nonumber \\ \end{aligned}$$

(15)

Now suppose $\alpha _{i-1}\ne \beta _{i-1}$ and $\alpha _{i-1}=(\alpha ,\alpha )^\pm =\beta _{i-1}$. Then

$$\begin{aligned} \displaystyle \sum _{\begin{array}{c} \alpha _{i-1}\ne \beta _{i-1}\\ \alpha _{i-1}=(\alpha ,\alpha )^\pm =\beta _{i-1} \end{array}}= & {} \displaystyle \sum _{\begin{array}{c} \alpha _j,\beta _j\in \mathcal {B}^j\\ (\alpha ,\alpha )^\pm \end{array}}M_\mathcal {B}((\alpha ,\alpha )^\pm ,\beta _{i-2})^2M_\mathcal {B}(\alpha _{i},(\alpha ,\alpha )^\pm )^2(d_{(\alpha ,\alpha )^\pm })^2\nonumber \\\le & {} \displaystyle \sum _{(\alpha ,\alpha )^\pm \in \mathcal {B}^{i-1}} {\text {jmp}}(\alpha )({\text {jmp}}(\alpha )+1)(d_{(\alpha ,\alpha )^\pm })^2. \end{aligned}$$

(16)

Summing equations (14), (15), and (16) gives part (2).

To prove (3), note that in this case

$$\begin{aligned} \#{\text {Hom}}(\mathcal {H}_i^i\uparrow G;\mathcal {B})=\sum _{\alpha _{i-1}\ne \beta _{i-1}}+\sum _{\alpha _{i-1}=\beta _{i-1}}, \end{aligned}$$

since $i-1$ is odd so $(\alpha , \alpha )^\pm \notin \mathcal {B}^{i-1}$. However, pairs of partitions of this form may be found at levels i and $i-2$.

First suppose $\alpha _{i-1}\ne \beta _{i-1}$. Then as in the proof of Lemma 7.17 they jointly determine $\alpha _i=(\alpha _i^1,\alpha _i^2)$. This means that they jointly determine at most two pairs of partitions (if $\alpha _i^1=\alpha _i^2$). Thus

$$\begin{aligned} \sum _{\alpha _{i-1}\ne \beta _{i-1}}\le & {} 2\sum _{\begin{array}{c} \beta _{i-2}\in \mathcal {B}^{i-2}\\ \alpha _{i-1}\ne \beta _{i-1}\in \mathcal {B}^{i-1} \end{array}}M_\mathcal {B}(\beta _{i-1},\beta _{i-2})M_\mathcal {B}(\alpha _{i-1},\beta _{i-2})d_{\alpha _{i-1}}d_{\beta _{i-1}} \nonumber \\= & {} 2\frac{\vert D_{i-1}\vert ^2}{\vert D_{i-2}\vert }-2\sum _{\beta _{i-1}\in \mathcal {B}^{i-1}}({\text {jmp}}(\beta _{i-1}^1)+{\text {jmp}}(\beta _{i-1}^2))(d_{\beta _{i-1}})^2, \end{aligned}$$

(17)

as in the proof of Lemma 7.17.

Now suppose $\alpha _{i-1}=\beta _{i-1}$. As before there are ${\text {jmp}}(\beta _{i-1}^1)$ ways to obtain $\beta _i^1$ and ${\text {jmp}}(\beta _{i-1}^2)$ ways to obtain $\beta _i^2$, but to account for when $\beta _i^1=\beta _i^2$, we overcount by multiplying by 2. The same holds for the number of ways to obtain $\alpha _{i-2}$ from $\beta _{i-1}$. Thus,

$$\begin{aligned} \sum _{\alpha _{i-1}\beta _{i-1}}\le & {} \sum _{\beta _{i-1}\in \mathcal {B}^{i-1}} 2({\text {jmp}}(\beta _{i-1}^1)+{\text {jmp}}(\beta _{i-1}^2))\nonumber \\&\times 2({\text {jmp}}(\beta _{i-1}^1)+{\text {jmp}}(\beta _{i-1}^2)+2)(d_{\beta _{i-1}})^2. \end{aligned}$$

(18)

Summing equations (17) and (18) gives part 3. $\square$

Combining Lemma 7.21 with Lemma 7.18 gives the following bound:

Corollary 7.22

$\#{\text {Hom}}(\mathcal {H}_i^n\uparrow G;\mathcal {B})\le \frac{20(i-1)}{n}\vert D_n\vert$.

Appendix 5: The General Linear Group

The SOV approach reduces Theorem 1.3 to counting the number of morphisms of the quivers of Fig. 19 into the Bratteli diagram of $Gl_n(q)$. In this section we use known results on the number of conjugacy classes and the multiplicities of representations of $Gl_n(q)$ to provide the bounds used in the proof of Theorem 1.3.

Theorem 7.23

For $\mathcal {H}_j^n$ the quiver of Fig. 19 and $\mathcal {B}$ the Bratteli diagram for the subgroup chain $Gl_n(q)>Gl_{n-1}(q)>\cdots >\{e\}$,

$$\begin{aligned} \begin{array}{ll}\dim A(\mathcal {H}_j^n\uparrow \mathcal {Q};\mathcal {B})\le 2^{2j-4}q^{j-2}\displaystyle \frac{q^{j-1}(q^j-1)}{q^{n-1}(q^n-1)}\vert Gl_{n}(q)\vert .\end{array} \end{aligned}$$

Proof

By Theorem 6.1, we see that $\# {\text {Hom}}(\mathcal {H}_j^n\uparrow \mathcal {Q};\mathcal {B})$ is equal to the sum

$$\begin{aligned}&\displaystyle \sum _{\alpha _i,\beta _i\in \mathcal {B}^i}M_\mathcal {B}(\beta _{n-1},\beta _{j-1})M_\mathcal {B}(\beta _{j-1},\beta _{j-2})M_\mathcal {B}(\alpha _{j},\beta _{j-1})M_\mathcal {B}(\alpha _{j},\alpha _{j-1})\\&\quad \quad \times M_\mathcal {B}(\alpha _{j-1},\beta _{j-2})d_{\alpha _{j-1}}d_{\beta _{n-1}}\\&\quad \le M_{\mathcal {B}}(Gl_{j-1},Gl_{j-2})^2\vert \hat{Gl}_{j-2}(q)\vert \displaystyle \sum _{\alpha _i,\beta _i\in \mathcal {B}^i}M_\mathcal {B}(\beta _{n-1},\beta _{j-1})M_\mathcal {B}(\alpha _{j},\beta _{j-1})\\&\quad \quad \times M_\mathcal {B}(\alpha _{j},\alpha _{j-1})d_{\alpha _{j-1}}d_{\beta _{n-1}}, \end{aligned}$$

for $M_\mathcal {B}(G_i,G_j)=M_\mathcal {B}(G_i(q),G_j(q)):=\max M_\mathcal {B}(\alpha _i,\alpha _j)$ over all $\alpha _i\in \mathcal {B}^i, \alpha _j\in \mathcal {B}^j$ and $\vert \hat{G}_i(q)\vert$ the number of conjugacy classes of $G_i(q).$ By Corollary 6.18,

$$\begin{aligned} \# {\text {Hom}}(\mathcal {H}_j^n\uparrow \mathcal {Q};\mathcal {B})\le & {} M_{\mathcal {B}}(Gl_{j-1}(q),Gl_{j-2}(q))^2\vert \hat{Gl}_{j-2}(q)\vert \langle D^{n-1}U^{n-j}DU^j\hat{0},\hat{0}\rangle \\= & {} M_{\mathcal {B}}(Gl_{j-1}(q),Gl_{j-2}(q))^2\vert \hat{Gl}_{j-2}(q)\vert \lambda _{j}\lambda _{n-1}\lambda _{n-2}\cdots \lambda _{1}\\= & {} M_{\mathcal {B}}(Gl_{j-1}(q),Gl_{j-2}(q))^2\vert \hat{Gl}_{j-2}(q)\vert \displaystyle \frac{\vert Gl_{j}(q)\vert }{\vert Gl_{j-1}(q)\vert }\vert Gl_{n-1}(q)\vert . \end{aligned}$$

By [32, Lemma 5.9], $M(Gl_j(q),Gl_{j-1}(q))\le 2^{j-1}$ and $\vert \hat{Gl}_{j}(q)\vert \le q^j$. Thus, since $\dim A(\mathcal {H}_j^n\uparrow \mathcal {Q};\mathcal {B})=\# {\text {Hom}}(\mathcal {H}_j^n\uparrow \mathcal {Q};\mathcal {B})$,

$$\begin{aligned} \begin{array}{ll} \dim A(\mathcal {H}_j^n\uparrow \mathcal {Q};\mathcal {B})&{}\le 2^{2j-4}q^{j-2}q^{j-1}(q^j-1)\vert Gl_{n-1}(q)\vert \\ &{}=2^{2j-4}q^{j-2}\displaystyle \frac{q^{j-1}(q^j-1)}{q^{n-1}(q^n-1)}\vert Gl_{n}(q)\vert \end{array} \end{aligned}$$

$\square$

Theorem 7.24

For $\mathcal {J}_j^n$ the quiver of Fig. 19,

$$\begin{aligned} \begin{array}{ll} \dim A(\mathcal {H}_j^n\uparrow \mathcal {Q};\mathcal {B})\le 2^{2j-4}q^{j-2}\displaystyle \frac{q^{j-1}(q^j-1)}{q^{n-1}(q^n-1)}\vert Gl_{n}(q)\vert . \end{array} \end{aligned}$$

Proof

$$\begin{aligned} \dim A(\mathcal {J}_j\uparrow \mathcal {Q};\mathcal {B})= & {} \sum _{\alpha _i,\beta _i\in \mathcal {B}^i}M_\mathcal {B}(\beta _{n-1},\beta _{j-1})\\&\quad \times M_\mathcal {B}(\alpha _{j},\beta _{j-1})M_\mathcal {B}(\alpha _{j},\alpha _{j-2})^2d_{\alpha _{j-2}}d_{\beta _{n-1}}\\\le & {} M_{\mathcal {B}}(Gl_{j}(q),Gl_{j-2}(q))\sum _{\alpha _i,\beta _i\in \mathcal {B}^i}M_\mathcal {B}(\beta _{n-1},\beta _{j-1})M_\mathcal {B}(\alpha _{j},\beta _{j-1})\\&\quad \times M_\mathcal {B}(\alpha _{j},\alpha _{j-2})d_{\alpha _{j-2}}d_{\beta _{n-1}}\\= & {} M_{\mathcal {B}}(Gl_{j}(q),Gl_{j-2}(q)) \langle D^{n-1}U^{n-j}DU^j\hat{0},\hat{0}\rangle \\= & {} M_{\mathcal {B}}(Gl_{j}(q),Gl_{j-2}(q))\frac{\vert Gl_{j}(q)\vert }{\vert Gl_{j-1}(q)\vert }\vert Gl_{n-1}(q)\vert . \end{aligned}$$

By [32, Lemma 5.9], $M(Gl_{j}(q),Gl_{j-2}(q))\le 2^{2j-3}q^{j-1}$. Thus,

$$\begin{aligned} \dim A(\mathcal {J}_j\uparrow \mathcal {Q};\mathcal {B})\le 2^{2j-3}q^{j-1}\frac{q^{j-1}(q^j-1)}{q^{n-1}(q^n-1)}\vert Gl_n(q)\vert . \end{aligned}$$

$\square$

1.1 Factoring Coset Representatives of $GL_n(\mathbb {F}_q)$

In this section we provide the set of coset representatives and their factorizations used in the proof of Theorem 1.3 by developing a correspondence between $Gl_n(q)/Gl_{n-1}(q)$ and the set $Z_n=\{\mathbf {z}=(x_1y_1,\dots , x_ny_n)\mid \mathbf {x},\mathbf {y}\in (\mathbb {F}_q)^n,\;y\cdot x=1\}.$

Define an action of $Gl_n(q)$ on $Z_n$ via $A.\mathbf {z}=(\tilde{x}_1\tilde{y}_1,\dots ,\tilde{x}_n\tilde{y}_n),$ for $\tilde{\mathbf {y}}=\mathbf {y}A^{-1},\tilde{\mathbf {x}}=(A\mathbf {x}^T)^T$. Note that the action of A preserves $y\cdot x$. For $\mathbf {1}=(0,\dots ,0,1)$, we show (Theorem 7.27) that

$$\begin{aligned} Z_n={\text {Orb}}(\mathbf {1}). \end{aligned}$$

Note that $Gl_{n-1}(q)$, viewed as a subgroup of $Gl_n(q)$, stabilizes $\mathbf {1},$ so the orbit-stabilizer theorem gives a bijection between $Z_n={\text {Orb}}(\mathbf {1})$ and $Gl_n(q)/Gl_{n-1}(q)$ through the correspondence

$$\begin{aligned} g.\mathbf {1}\longleftrightarrow gGl_{n-1}(q). \end{aligned}$$

(19)

Thus, writing $\mathbf {z}=A_1\cdots A_m.\mathbf {1}$ for each $\mathbf {z}\in Z_n$ gives a factorization of the corresponding coset representative. We find a factorization in which each matrix $A_i=A\bigoplus I_{n-2}$ for $A\in Gl_2(q)$.

Lemma 7.25

Suppose $\mathbf {z}=(x_1y_1,x_2y_2)\in Z_2$ with $x_1y_1+x_2y_2\ne 0$. Then there exists a matrix $A\in Gl_2(q)$ and $y_2',x_2'\in \mathbb {F}_q^\times$ such that

$$\begin{aligned} A.\mathbf {z}=(0,x_2'y_2'). \end{aligned}$$

Proof

Case 1: $x_1=0$. Let $A=\begin{pmatrix} 1&{}0\\ \frac{y_1}{y_2}&{}1 \end{pmatrix}$. Note that for all possible choices of $\mathbf {z}\in Z_2$, there are q possibilities for A.

Case 2: $x_1\ne 0$, $y_1=0$. Let $A=\begin{pmatrix} 1&{}\frac{-x_1}{x_2}\\ 0&{}1 \end{pmatrix}$. Note there are $q-1$ possibilities for A.

Case 3: $x_1\ne 0$, $y_1\ne 0$. Let $A=\begin{pmatrix} \frac{-x_2}{x_1}&{}1\\ 1&{}\frac{y_2}{y_1} \end{pmatrix}$. Note there are $q^2$ possibilities for A. Note further that for $z_1:=x_1y_1$ and $z_2:=x_2y_2$ fixed and nonzero,

$$\begin{aligned} A=\begin{pmatrix} \frac{-x_2}{x_1}&{}1\\ 1&{}\frac{z_2}{z_1}\frac{x_1}{x_2} \end{pmatrix}, \end{aligned}$$

and there are $q-1$ possibilities for A. $\square$

We use Lemma 7.25 to systematically write $\mathbf {z}\in Z_n$ in form

$$\begin{aligned} \mathbf {z}=A_1A_2\cdots A_k.\mathbf {1}, \end{aligned}$$

with $A_i\in Gl_n(q)$. Recall that p is the characteristic of $\mathbb {F}_q$.

Proposition 7.26

Let $\tilde{\mathbf {z}}\in Z_n$. Then there is a permutation matrix $\pi \in Gl_n(q)$, $b\in \mathbb {F}_q^\times$, and $i\ge 1$ such that $\pi .\tilde{\mathbf {z}}=\mathbf {z}$ with:

• (i) $z_1+\dots +z_j\ne 0$ for all $i\le j\le n$,

• (ii) $z_1=\dots =z_i=b$,

• (iii) $p\vert (i-1)$.

Proof

Let $\tilde{\mathbf {z}}=(\tilde{z}_1,\dots ,\tilde{z}_n)\in Z_n$. Note that $\tilde{z}_1+\cdots +\tilde{z}_n=1\ne 0$. Let j be an index (if it exists) such that $\tilde{z}_1+\cdots \tilde{z}_n-\tilde{z}_j\ne 0$. Note that for a permutation matrix $\pi$,

$$\begin{aligned} \pi .\tilde{\mathbf {z}}=(\tilde{z}_{\pi (1)},\dots , \tilde{z}_{\pi (n)}). \end{aligned}$$

Permute $\tilde{\mathbf {z}}$ to make $\tilde{z}_j$ the last entry, then delete $\tilde{z}_j$ to produce a vector of length $n-1$. Repeat until no such index exists, and let i be the length of the resultant vector, $\mathbf {z}$. Then clearly $z_1+\cdots +z_j\ne 0$ for all $i\le j\le n$. Further, $z_1+\cdots +z_i-z_k=0$ for all $1\le k\le i$; in particular, $z_1=\cdots =z_i=b\in \mathbb {F}_q^\times$. Finally note that $z_1+\cdots +z_{i-1}=(i-1)b=0$ and so $p\vert (i-1)$. $\square$

In light of Proposition 7.26, let

$$\begin{aligned} S_i(n)=\{\mathbf {z}\in Z_n\vert \;\mathbf {z}\; \text {satisfies (i) and (ii) of Proposition E.4} \}. \end{aligned}$$

Theorem 7.27

For $p\ne 2$ and $\mathbf {z}\in S_i(n)$, there exist invertible matrices $u_j,u_j', v_j, t_j\in Gl_j(q)\cap {\text {Centralizer}}(Gl_{j-2}(q)),$ such that

$$\begin{aligned} v_n\cdots v_{i+1}u_{i}\cdots u_{2p+1}t_{2p}u_{2p+1}'u_{2p-1}\cdots u_{p+1}t_p(u_{p+1}')u_{p-1}\cdots u_2.\mathbf {z}=\mathbf {1}. \end{aligned}$$

Proof

Let $\mathbf {z}\in S_i(n)$ and let $i>p$. Note that $\mathbf {z}=\begin{pmatrix} b,\dots ,b,z_{i+1}, \dots ,z_n \end{pmatrix},$ and since $z_1+z_2=2b\ne 0$, by Lemma 7.25, there is a matrix $A\in Gl_2(q)$ such that $A.(z_1,z_2)=\begin{pmatrix}0,x_2'y_2'\end{pmatrix}$ with $y_2'x_2'=2b$. Let $u_2= A\bigoplus I_{n-2}\in Gl_n(q).$ Then

$$\begin{aligned} u_2.\mathbf {z}=\begin{pmatrix}0,2b,b,\dots ,b,z_{i+1},\dots ,z_n\end{pmatrix}. \end{aligned}$$

Repeat this process, defining matrices $u_3,u_4,\dots ,u_{p-1}$ (i.e., find the matrix A guaranteed by Lemma 7.25, and let $u_j=I_{j-2}\bigoplus A\bigoplus I_{n-j}$). Note that

$$\begin{aligned} u_{p-1}\cdots u_3u_2.\mathbf {z}= \begin{pmatrix} 0,\dots ,0,(p-1)b,b,b,b,\dots ,b,z_{i+1},\dots ,z_n \end{pmatrix}. \end{aligned}$$

Since $z_{p-1}+z_p=pb=0$, we cannot use Lemma 7.25. Instead, define $(u_{p+1}')$ as above and let $t_p$ be the permutation matrix of the transposition $(p-1\; p)$. Then

$$\begin{aligned} t_pu_{p+1}'u_{p-1}\cdots u_2.\mathbf {z}=\begin{pmatrix} 0,\dots ,0,(p-1)b,2b,b,b,\dots ,b,z_{i+1},\dots ,z_n \end{pmatrix}, \end{aligned}$$

and since now $z_{p-1}+z_p\ne 0$, define $u_{p+1}$ as before so that

$$\begin{aligned} u_{p+1}t_pu_{p+1}'u_{p-1}\cdots u_2.\mathbf {z} =\begin{pmatrix} 0,\dots ,0,0,(p+1)b=b,b,b,\dots ,b,z_{i+1},\dots ,z_n \end{pmatrix} \end{aligned}$$

Repeat this process through definition of the matrix $u_i$, so that

$$\begin{aligned} u_i\cdots u_2.\mathbf {z}=\begin{pmatrix} 0,\dots ,0,z_1+\cdots +z_i,z_{i+1},\dots ,z_n \end{pmatrix}. \end{aligned}$$

Since $z_1+\cdots +z_j\ne 0$ for all $i\le j\le n$, we use Lemma 7.25 to find the appropriate 2x2 matrix $A_j$ so that for $v_j=I_{j-2}\bigoplus A_j\bigoplus I_{n-j}$,

$$\begin{aligned} v_n\cdots v_{i+1}u_i\cdots u_2.\mathbf {z}= \begin{pmatrix} 0,\dots ,0,z_1+\cdots +z_n \end{pmatrix}=\begin{pmatrix} 0,\dots ,0,1 \end{pmatrix}. \end{aligned}$$

For $i<p$ analogous arguments apply without needing the matrices $t_p$. $\square$

Remark 7.28

By Lemma 7.25, there are $(q-1)$ possibilities for each $u_j$ and $q^2$ possibilities for each $v_j$.

By Proposition 7.26,

$$\begin{aligned} X_n=\bigcup _{\pi \in S_n}\bigcup _{\begin{array}{c} 1\le i\le n \\ p\vert (i-1) \end{array}}\pi S_i(n) \end{aligned}$$

and so by Expression 19 a complete set of coset representatives for $Gl_n(q)/Gl_{n-1}(q)$ is contained in $\{\pi s_i\vert \;1\le i\le n, p\mid (i-1), s_i\in S_i(n)\},$ with each $s_i$ of form:

$$\begin{aligned} s_i=u_2\cdots u_{p-1}u_{p+1}'t_pu_{p+1}\cdots u_iv_{i+1}\cdots v_n. \end{aligned}$$

Finally, we note that similar results hold in the $p=2$ case.

Theorem 7.29

For $p=2$, $i\ge 3$ odd, $(\mathbf {y},\mathbf {x})\in S_i(n)$, there exist invertible matrices

$$\begin{aligned} a_j,b_j,c_j,v_j\in Gl_j(q)\cap {\text {Centralizer}}(Gl_{j-2}(q)) \end{aligned}$$

such that

$$\begin{aligned} \mathbf {z}=a_3b_2c_3\cdots a_ib_{i-1}c_iv_{i+1}\cdots v_n.\mathbf {1}. \end{aligned}$$

Note that there are $(q-1)$ choices for $a_j$ and $b_j$, that $c_j$ is completely determined by $a_j$ and $b_j$, and that there are $q^2$ choices for $v_j$.