Unipotent Hecke algebras: the structure, representation theory, and combinatorics

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 26 March 2015

This is an excerpt of the PhD thesis Unipotent Hecke algebras: the structure, representation theory, and combinatorics by F. Nathaniel Edgar Thiem.

The representation theory of the Yokonuma algebra

General type

Let $1:U\to {ℂ}^{*}$ be the trivial character, and let $$\begin{array}{cc}{\displaystyle e_1=\frac{1}{\mid U\mid }\sum _{u\in U}u\in ℂG}& \text{(6.1)}\end{array}$$ be the idempotent so that ${\text{Ind}}_U^G\left(1\right)\cong ℂG{e}_1\text{.}$ Then the Yokonuma algebra $${\mathscr{H}}_1={\text{End}}_{ℂG}\left(ℂG{e}_1\right)\cong e_1ℂG{e}_1$$ has a basis $$\left\{e_{1}v{e}_1\hspace{0.17em}\right|\hspace{0.17em}v\in N\},\text{indexed by}N=⟨{\xi }_i,h\hspace{0.17em}|\hspace{0.17em}i=1,2,\dots ,\ell ,\hspace{0.17em}h\in T⟩,$$ where ${\xi }_i=w_i\left(1\right)=x_i\left(1\right)x_{-{\alpha }_i}(-1)x_i\left(1\right),$ and $T=⟨h_H\left(t\right)\hspace{0.17em}|\hspace{0.17em}H\in {𝔥}_{\mathbb{Z}}⟩\text{.}$ Recall that $h_i\left(t\right)=h_{H_{{\alpha }_i}}\left(t\right)\text{.}$

The map $$\begin{array}{ccc}ℂT& ⟶& {\mathscr{H}}_1\\ h& ⟼& e_{1}h{e}_1\end{array}$$ is an injective algebra homomorphism (see Chapter 3 (E3)). For $v\in N,$ write $$T_v=e_{1}v{e}_1\in {\mathscr{H}}_1,\text{with}{T}_i=T_{{\xi }_i},\text{and}h=T_h,h\in T\text{.}$$

(Yokonuma). ${\mathscr{H}}_1$ is generated by $T_i,$ $i=1,\dots ,\ell ,$ and $h\in T$ with relations $$\begin{array}{rcl}{\displaystyle T_{i}h}& =& {\displaystyle s_i\left(h\right)T_i,}\\ {\displaystyle T_i^2}& =& {\displaystyle q^{-1}{h}_i(-1)+q^{-1}\sum _{t\in {𝔽}_q^{*}}{h}_i\left(t\right)T_i,}\\ {\displaystyle \underset{\underset{m_{ij}}{⏟}}{T_i{T}_j{T}_i\cdots }}& =& {\displaystyle \underset{\underset{m_{ij}}{⏟}}{T_j{T}_i{T}_j\cdots },\text{where}\hspace{0.17em}{m}_{ij}\hspace{0.17em}\text{is the order of}\hspace{0.17em}{s}_i{s}_j\hspace{0.17em}\text{in}\hspace{0.17em}W,}\\ {\displaystyle T_h{T}_k}& =& {\displaystyle T_{hk},h,k\in T\text{.}}\end{array}$$

A reduction theorem

Let $\hat{T}$ index the irreducible $ℂT\text{-modules.}$ Since $T$ is abelian, all the irreducible modules are one-dimensional, and we may identify the label $\gamma \in \hat{T}$ of $V^{\gamma }=ℂ\text{-span}\left\{v_{\gamma }\right\}$ with the homomorphism $\gamma :T\to {ℂ}^{*}$ given by $$h{v}_{\gamma }=\gamma \left(h\right)v_{\gamma },\text{for}\hspace{0.17em}h\in T\text{.}$$ Suppose $V$ is an ${\mathscr{H}}_1\text{-module.}$ As a $T\text{-module}$ $$V=\underset{\gamma \in \hat{T}}{⨁}{V}_{\gamma },\text{where}\hspace{0.17em}{V}_{\gamma }=\{v\in V\hspace{0.17em}|\hspace{0.17em}hv=\gamma \left(h\right)v,\hspace{0.17em}h\in T\}\text{.}$$

Note that $$\begin{array}{cc}{\displaystyle {\mathscr{H}}_1=\underset{\gamma \in \hat{T}}{⨁}{\mathscr{H}}_1{\tau }_{\gamma },\text{where}{\tau }_{\gamma }=\frac{1}{\mid T\mid }\sum _{h\in T}\gamma \left(h^{-1}\right)h\text{.}}& \text{(6.3)}\end{array}$$ Recall the surjection $\pi :N\to W$ of (3.3). Use this map to identify $$\begin{array}{cc}{\displaystyle w=s_{i_1}{s}_{i_2}\cdots s_{i_r}\in W⟷w={\xi }_{i_1}{\xi }_{i_2}\cdots {\xi }_{i_r}\in N,\text{for}\hspace{0.17em}r\hspace{0.17em}\text{minimal.}}& \text{(6.4)}\end{array}$$ Thus, the $W\text{-action}$ on $\hat{T}$ $$\left(w\gamma \right)\left(h\right)=\gamma \left(w^{-1}\left(h\right)\right),\text{for}\hspace{0.17em}w\in W,\hspace{0.17em}h\in T\text{, and}\hspace{0.17em}\gamma \in \hat{T},$$ implies that for $\gamma \in \hat{T},$ $v_{\gamma }\in V_{\gamma },$ $$h{T}_w{v}_{\gamma }=T_w{w}^{-1}\left(h\right)v_{\gamma }=\gamma \left(w^{-1}h\right)T_w{v}_{\gamma }\text{for all}\hspace{0.17em}h\in T\text{.}$$ Thus, $$\begin{array}{cc}{\displaystyle T_w\left(V_{\gamma }\right)\subseteq V_{w\gamma }\text{.}}& \text{(6.5)}\end{array}$$ Let $$W_{\gamma }=\{w\in W\hspace{0.17em}|\hspace{0.17em}w\left(\gamma \right)=\gamma \}$$ and $$\begin{array}{cc}{\displaystyle {𝒯}_{\gamma }={\tau }_{\gamma }{\mathscr{H}}_1{\tau }_{\gamma }=ℂ\text{-span}\left\{{\tau }_{\gamma }{T}_w{\tau }_{\gamma }\hspace{0.17em}\right|\hspace{0.17em}w\in W_{\gamma }\}\text{.}}& \text{(6.6)}\end{array}$$

Remarks 1. Since the identity of ${\mathscr{H}}_1$ is $e_1$ and the identity of ${𝒯}_{\gamma }$ is ${\tau }_{\gamma },$ the algebra ${𝒯}_{\gamma }$ is not a subalgebra of ${\mathscr{H}}_1\text{.}$ In fact, ${𝒯}_{\gamma }\cong {\text{End}}_{{\mathscr{H}}_1}\left({\text{Ind}}_{ℂT}^{{\mathscr{H}}_1}\left(\gamma \right)\right)\text{.}$ 2. If $V$ is an ${\mathscr{H}}_1\text{-module,}$ then $${\tau }_{\gamma }V=V_{\gamma },\text{and so}{𝒯}_{\gamma }V={𝒯}_{\gamma }{V}_{\gamma }\text{.}$$ In particular, $V_{\gamma }$ is an ${𝒯}_{\gamma }\text{-module.}$

Let $V$ be an irreducible ${\mathscr{H}}_1\text{-module}$ such that $V_{\gamma }\ne 0\text{.}$ Then (a) $V_{\gamma }$ is an irreducible ${𝒯}_{\gamma }\text{-module;}$ (b) if $M$ is an irreducible ${𝒯}_{\gamma }\text{-module,}$ then ${\mathscr{H}}_1{\otimes }_{{𝒯}_{\gamma }}M$ is an irreducible ${\mathscr{H}}_1\text{-module;}$ (c) the map $$\begin{array}{ccccc}{\text{Ind}}_{{𝒯}_{\gamma }}^{{\mathscr{H}}_1}\left(V_{\gamma }\right)& =& {\mathscr{H}}_1{\otimes }_{{𝒯}_{\gamma }}{V}_{\gamma }& \stackrel{\sim }{⟶}& V\\ & & T_w\otimes v_{\gamma }& ⟼& T_w{v}_{\gamma }\end{array}$$ is an ${\mathscr{H}}_1\text{-module}$ isomorphism.

Let $\hat{T}/W$ be the set of $W\text{-orbits}$ in $\hat{T}\text{.}$ Identify $\hat{T}/W$ with a set of orbit representatives, and for $\gamma \in \hat{T}/W,$ let ${\hat{T}}_{\gamma }$ index the irreducible modules of ${𝒯}_{\gamma }\text{.}$

The map $$\begin{array}{cc}\left\{\text{Irreducible}\hspace{0.17em}{\mathscr{H}}_1\text{-modules}\right\}& ⟷& \{\text{Pairs}\hspace{0.17em}(\gamma ,\lambda ),\gamma \in \hat{T}/W,\lambda \in {\widehat{𝒯}}_{\gamma }\}\\ {\text{Ind}}_{{𝒯}_{\gamma }}^{{\mathscr{H}}_1}\left({𝒯}_{\gamma }^{\lambda }\right)& \leftrightarrow & (\gamma ,\lambda )\end{array}$$ is a bijection.

The algebras ${𝒯}_{\gamma }$

Let $B=UT$ be a Borel subgroup of $G\text{.}$ Since $$\begin{array}{ccc}B& ⟶& T\\ uh& ⟼& h\end{array}$$ is a surjective homomorphism, $\gamma :T\to {ℂ}^{*}$ extends to a linear character $\gamma$ of $B$ given by $\gamma \left(uh\right)=\gamma \left(h\right)\text{.}$ Note that $$e_1{\tau }_{\gamma }=\frac{1}{\mid B\mid }\sum _{b\in B}\gamma \left(b^{-1}\right)b\text{(}{e}_1\hspace{0.17em}\text{as in (6.1)).}$$ Thus, $${𝒯}_{\gamma }={\tau }_{\gamma }{e}_1ℂG{e}_1{\tau }_{\gamma }\cong {\text{End}}_{ℂG}\left({\text{Ind}}_B^G\left(\gamma \right)\right)=\mathscr{H}(G,B,\gamma ),$$ where if $\gamma$ is the trivial character $1_B,$ then $\mathscr{H}(G,B,1_B)$ is the Iwahori-Hecke algebra.

Let $\gamma \in \hat{T}$ be such that $W_{\gamma }$ is generated by simple reflections. Then ${𝒯}_{\gamma }$ is presented by generators $\left\{T_i\hspace{0.17em}\right|\hspace{0.17em}{s}_i\in W_{\gamma }\}$ with relations $$\begin{array}{rcl}{\displaystyle T_i^2}& =& {\displaystyle \{\begin{array}{ll}{q}^{-1}+q^{-1}(q-1)T_i,& \text{if}\hspace{0.17em}\gamma \left(h_{{\alpha }_i}\left(t\right)\right)=1\hspace{0.17em}\text{for all}\hspace{0.17em}t\in {𝔽}_q^{*},\\ \gamma \left(h_{{\alpha }_i}(-1)\right)q^{-1},& \text{otherwise,}\end{array}}\\ {\displaystyle \underset{\underset{m_{ij}}{⏟}}{T_i{T}_j{T}_i\cdots }}& =& {\displaystyle \underset{\underset{m_{ij}}{⏟}}{T_j{T}_i{T}_j\cdots },\text{where}\hspace{0.17em}{m}_{ij}\text{is the order of}\hspace{0.17em}{s}_i{s}_j\hspace{0.17em}\text{in}\hspace{0.17em}W\text{.}}\end{array}$$

The $G={GL}_n\left({𝔽}_q\right)$ case

The Yokonuma algebra and the Iwahori-Hecke algebra

If ${\psi }_{\mu }=1$ is the trivial character of $U,$ then $\mu =\left(1^n\right)\text{.}$ Let $T_i=e_1{s}_i{e}_1$ and recall that $h_{{\epsilon }_j}\left(t\right)={\text{Id}}_{j-1}\oplus \left(t\right)\oplus {\text{Id}}_{n-j}\text{.}$ In the case $G={GL}_n\left({𝔽}_q\right),$ ${\mathscr{H}}_1$ has generators $T_i,$ $h_{{\epsilon }_j}\left(t\right),$ for $1\le i<n,$ $1\le j\le n$ and $t\in {𝔽}_q^{*}$ with relations $$\begin{array}{rcl}{\displaystyle T_i^2}& =& {\displaystyle q^{-1}+q^{-1}{h}_{{\epsilon }_i}(-1)\sum _{t\in {𝔽}_q^{*}}{h}_{{\epsilon }_i}\left(t\right)h_{{\epsilon }_{i+1}}\left(t^{-1}\right)T_i,}\\ {\displaystyle T_i{T}_{i+1}{T}_i}& =& {\displaystyle T_{i+1}{T}_i{T}_{i+1},T_i{T}_j=T_j{T}_i,\mid i-j\mid >1,}\\ {\displaystyle h_{{\epsilon }_j}\left(t\right)T_i}& =& {\displaystyle T_i{h}_{{\epsilon }_{j\prime }}\left(t\right),\text{where}\hspace{0.17em}j\prime =s_i\left(j\right),}\\ {\displaystyle h_{{\epsilon }_i}\left(a\right)h_{{\epsilon }_i}\left(b\right)}& =& {\displaystyle h_{{\epsilon }_i}\left(ab\right),h_{{\epsilon }_i}\left(a\right)h_{{\epsilon }_j}\left(b\right)=h_{{\epsilon }_j}\left(b\right)h_{{\epsilon }_i}\left(a\right),a,b\in {𝔽}_q^{*}\text{.}}\end{array}$$

The Yokonuma algebra has a decomposition $${\mathscr{H}}_1=\underset{\gamma \in \hat{T}}{⨁}{\mathscr{H}}_1{\tau }_{\gamma },\text{where}{\tau }_{\gamma }=\frac{1}{\mid T\mid }\sum _{h\in T}\gamma \left(h^{-1}\right)h\text{.}$$

Fix a total ordering $\le$ on the set ${\widehat{𝔽}}_q^{*}=\{{\phi }_1,{\phi }_2,\dots ,{\phi }_{q-1}\}$ of linear characters of ${𝔽}_q^{*},$ such that ${\phi }_1:{𝔽}_q^{*}\to {ℂ}^{*}$ is the trivial character. Suppose $\gamma \in \hat{T}\text{.}$ Since $⟨h_{{\epsilon }_i}\left(t\right)\hspace{0.17em}|\hspace{0.17em}t\in {𝔽}_q^{*}⟩\cong {𝔽}_q^{*},$ $$\begin{array}{rcll}{\displaystyle \gamma \left(h\right)}& =& {\displaystyle \gamma \left(h_{{\epsilon }_1}\left(h_1\right)\right)\gamma \left(h_{{\epsilon }_2}\left(h_2\right)\right)\cdots \gamma \left(h_{{\epsilon }_n}\left(h_n\right)\right),}& {\displaystyle \text{where}\hspace{0.17em}h=\text{diag}(h_1,h_2,\dots ,h_n)\in T,}\\ & =& {\displaystyle {\gamma }_1\left(h_1\right){\gamma }_2\left(h_2\right)\cdots {\gamma }_n\left(h_n\right),}& {\displaystyle \text{where}\hspace{0.17em}{\gamma }_i\in {\widehat{𝔽}}_q^{*}\text{.}}\end{array}$$ Thus, every $\gamma \in \hat{T}$ can be written $\gamma ={\phi }_{i_1}\otimes {\phi }_{i_2}\otimes \cdots \otimes {\phi }_{i_n}\text{.}$

Let $\nu =({\nu }_1,{\nu }_2,\dots ,{\nu }_n)\models n$ with ${\nu }_i\ge 0\text{.}$ Write $$\begin{array}{cc}{\displaystyle {\gamma }_{\nu }=\underset{\underset{{\nu }_1\hspace{0.17em}\text{terms}}{⏟}}{{\phi }_1\otimes \cdots \otimes {\phi }_1}\otimes \underset{\underset{{\nu }_2\hspace{0.17em}\text{terms}}{⏟}}{{\phi }_2\otimes \cdots \otimes {\phi }_2}\otimes \cdots \otimes \underset{\underset{{\nu }_n\hspace{0.17em}\text{terms}}{⏟}}{{\phi }_n\otimes \cdots \otimes {\phi }_n}\text{and}{\tau }_{\nu }={\tau }_{{\gamma }_{\nu }}\text{.}}& \text{(6.7)}\end{array}$$ Note that the ${\gamma }_{\nu }$ are orbit representatives of the $W\text{-action}$ in $\hat{T}\text{.}$ Let $$\begin{array}{rcl}{\displaystyle W_{\nu }}& =& {\displaystyle \{w\in W\hspace{0.17em}|\hspace{0.17em}w\left({\gamma }_{\nu }\right)={\gamma }_{\nu }\}=W_{{\nu }_1}\oplus W_{{\nu }_2}\oplus \cdots \oplus W_{{\nu }_n},}\\ {\displaystyle {𝒯}_{\nu }}& =& {\displaystyle {\tau }_{\nu }{\mathscr{H}}_1{\tau }_{\nu }=ℂ\text{-span}\left\{{\tau }_{\nu }{T}_w{\tau }_{\nu }\hspace{0.17em}\right|\hspace{0.17em}w\in W_{\nu }\}\text{.}}\end{array}$$ Note that $W_{\nu }$ is generated by its simple reflections.

If $s_i$ is in the $k\text{th}$ factor of $W_{\nu }=W_{{\nu }_1}\oplus W_{{\nu }_2}\oplus \cdots \oplus W_{{\nu }_n},$ then write $$s_i\in W_{{\nu }_k}\text{.}$$

Let $\nu \models n$ and ${}^{\nu }{T}_w={\tau }_{\nu }{T}_w{\tau }_{\nu }$ for $w\in W_{\nu }\text{.}$ Then ${𝒯}_{\nu }$ is presented by generators $$\left\{{}^{\nu }{T}_i\hspace{0.17em}\right|\hspace{0.17em}{s}_i\in W_{\nu }\}$$ with relations $$\begin{array}{rcl}{\displaystyle {}^{\nu }{T}_i^2}& =& {\displaystyle q^{-1}+{\phi }_k(-1)q^{-1}{(q-1)}^{\nu }{T}_i,\text{for}\hspace{0.17em}{s}_i\in W_{{\nu }_k}}\\ {\displaystyle {}^{\nu }{T}_i{}^{\nu }{T}_{i+1}{}^{\nu }{T}_i}& =& {\displaystyle {}^{\nu }{T}_{i+1}{}^{\nu }{T}_i{}^{\nu }{T}_{i+1}}\\ {\displaystyle {}^{\nu }{T}_i{}^{\nu }{T}_j}& =& {\displaystyle {}^{\nu }{T}_j{}^{\nu }{T}_i,\text{for}\hspace{0.17em}\mid i-j\mid >1\text{.}}\end{array}$$

The Iwahori-Hecke algebra $𝒯\subseteq {\mathscr{H}}_1$ is the algebra $${𝒯}_n={\mathscr{H}}_{\left(n\right)},\text{(recall}\hspace{0.17em}{\phi }_1\hspace{0.17em}\text{is the trivial character).}$$ In ${𝒯}_n,$ write $$I_i={\tau }_{\left(n\right)}{T}_i{\tau }_{\left(n\right)}\text{.}$$

Let $\nu =({\nu }_1,\dots ,{\nu }_n)\models n\text{.}$ Index the generators $I_i$ of ${𝒯}_{{\nu }_k}$ by $\left\{i\hspace{0.17em}\right|\hspace{0.17em}{s}_i\in W_{{\nu }_k}\}\text{.}$ Then the map $$\begin{array}{ccc}{𝒯}_{\nu }& \stackrel{\sim }{⟶}& {𝒯}_{{\nu }_1}\otimes {𝒯}_{{\nu }_2}\otimes \cdots \otimes {𝒯}_{{\nu }_n}\\ {}^{\nu }{T}_i& ⟼& {\phi }_k(-1)\underset{\underset{k-1\hspace{0.17em}\text{terms}}{⏟}}{1\otimes \cdots \otimes 1}\otimes I_i\otimes \underset{\underset{n-k\hspace{0.17em}\text{terms}}{⏟}}{1\otimes \cdots \otimes 1},\text{for}\hspace{0.17em}{s}_i\in W_{{\nu }_k},\end{array}$$ is an algebra isomorphism.

The representation theory of ${𝒯}_{\nu }$

By Corollary 6.6, understanding the representation theory of ${𝒯}_{\nu }$ is the same as understanding the representation theory of the Iwahori-Hecke algebras ${𝒯}_{{\nu }_i}\text{.}$

Let $\mu ⊢n$ be a partition. A standard tableau of shape $\mu$ is a column strict tableau of shape $\mu$ and weight $\left(1^n\right)$ (i.e. every number between from $1$ to $n$ appears exactly once). Suppose $P$ is a standard tableau of shape $\mu \text{.}$ Let $$\begin{array}{cc}{\displaystyle \begin{array}{rcl}{\displaystyle c_P\left(i\right)}& =& {\displaystyle \text{the content of the box containing}\hspace{0.17em}i,}\\ {\displaystyle C_P\left(i\right)}& =& {\displaystyle \frac{q-1}{1-q^{c_P\left(i\right)-c_P(i+1)}}\text{.}}\end{array}}& \text{(6.8)}\end{array}$$

([Hoe1974, Ram1997, Wen1988]). Let $G={GL}_n\left({𝔽}_q\right)\text{.}$ Then (a) The irreducible ${𝒯}_n\text{-modules}$ ${𝒯}_n^{\mu }$ are indexed by partitions $\mu ⊢n,$ (b) $\text{dim}\left({𝒯}_n^{\mu }\right)=\text{Card}\left\{\text{standard tableau}\hspace{0.17em}P\hspace{0.17em}\right|\hspace{0.17em}\text{sh}\left(P\right)=\mu \},$ (c) Let ${𝒯}_n^{\mu }=ℂ\text{-span}\left\{v_P\hspace{0.17em}\right|\hspace{0.17em}\text{sh}\left(P\right)=\mu \text{, wt}\left(P\right)=\left(1^n\right)\}\text{.}$ Then $$I_i{v}_P=q^{-1}{C}_P\left(i\right)v_P+q^{-1}(1+C_P\left(i\right))v_{s_{i}P}$$ where $v_{s_{i}P}=0$ if $s_{i}P$ is not a column strict tableau.

Recall from Chapter 5, an ${\widehat{𝔽}}_q^{*}\text{-partition}$ $\lambda =({\lambda }^{\left({\phi }_1\right)},{\lambda }^{\left({\phi }_2\right)},\dots ,{\lambda }^{\left({\phi }_{q-1}\right)})$ is a sequence of partitions indexed by $\widehat{𝔽}q*\text{.}$ Let $$\mid \lambda \mid =\mid {\lambda }^{\left({\phi }_1\right)}\mid +\mid {\lambda }^{\left({\phi }_2\right)}\mid +\cdots +\mid {\lambda }^{\left({\phi }_{q-1}\right)}\mid =\text{total number of boxes.}$$

Let $\nu =({\nu }_1,{\nu }_2,\dots ,{\nu }_n)\models n$ with ${\nu }_i\ge 0$ ${\gamma }_{\nu }$ as in (6.7). A standard $\nu \text{-tableau}$ $$Q=(Q^{\left({\phi }_1\right)},Q^{\left({\phi }_2\right)},\dots ,Q^{\left({\phi }_{q-1}\right)})$$ of shape $\lambda$ is a column strict filling of $\lambda$ by the numbers $1,2,3,\dots ,n$ such that (a) each number appears exactly once, (b) $j\in Q^{\left({\phi }_k\right)}$ if $s_j\in W_{{\nu }_k}\text{.}$ Let $$\begin{array}{rcll}{\displaystyle {\widehat{𝒯}}_{\nu }^{\lambda }}& =& {\displaystyle \left\{\text{standard}\hspace{0.17em}\nu \text{-tableau}\hspace{0.17em}Q\hspace{0.17em}\right|\hspace{0.17em}\text{sh}\left(Q\right)=\lambda \},}& \text{(6.9)}\\ {\displaystyle {\widehat{𝒯}}_{\nu }}& =& {\displaystyle \left\{{\widehat{𝔽}}_q^{*}\text{-partition}\hspace{0.17em}\lambda \hspace{0.17em}\right|\hspace{0.17em}{\widehat{𝒯}}_{\nu }^{\lambda }\ne \varnothing \}}\\ {\displaystyle }& =& {\displaystyle \left\{{\widehat{𝔽}}_q^{*}\text{-partition}\hspace{0.17em}\lambda \hspace{0.17em}\right|\hspace{0.17em}\mid {\lambda }^{\left({\phi }_i\right)}\mid =\mid {\nu }_i\mid \}\text{.}}& \text{(6.10)}\end{array}$$

Example. If $\nu =(3,0,1,0),$ then ${\gamma }_{\nu }={\phi }_1\otimes {\phi }_1\otimes {\phi }_1\otimes {\phi }_3\text{.}$ The set ${\widehat{𝒯}}_{\nu }$ is $$\{({\begin{array}{c} \end{array}}^{\left({\phi }_1\right)},{\varnothing }^{\left({\phi }_2\right)},{\begin{array}{c} \end{array}}^{\left({\phi }_3\right)},{\varnothing }^{\left({\phi }_4\right)}),({\begin{array}{c} \end{array}}^{\left({\phi }_1\right)},{\varnothing }^{\left({\phi }_2\right)},{\begin{array}{c} \end{array}}^{\left({\phi }_3\right)},{\varnothing }^{\left({\phi }_4\right)}),({\begin{array}{c} \end{array}}^{\left({\phi }_1\right)},{\varnothing }^{\left({\phi }_2\right)},{\begin{array}{c} \end{array}}^{\left({\phi }_3\right)},{\varnothing }^{\left({\phi }_4\right)})\},$$ and, for example, $${\widehat{𝒯}}_{\nu }^{({\begin{array}{c} \end{array}}^{\left({\phi }_1\right)},{\begin{array}{c} \end{array}}^{\left({\phi }_3\right)})}=\{({\begin{array}{c} 1 2 3 \end{array}}^{\left({\phi }_1\right)},{\varnothing }^{\left({\phi }_2\right)},{\begin{array}{c} 4 \end{array}}^{\left({\phi }_3\right)},{\varnothing }^{\left({\phi }_4\right)}),({\begin{array}{c} 1 3 2 \end{array}}^{\left({\phi }_1\right)},{\varnothing }^{\left({\phi }_2\right)},{\begin{array}{c} 4 \end{array}}^{\left({\phi }_3\right)},{\varnothing }^{\left({\phi }_4\right)})\}$$

Let $Q\in {\widehat{𝒯}}_{\nu }^{\lambda }\text{.}$ Suppose $s_j\in W_{{\nu }_k}$ so that $j\in Q^{\left({\phi }_k\right)}\text{.}$ Write $$\begin{array}{rcll}{\displaystyle Q_j}& =& {\displaystyle {\phi }_k,}& \text{(6.11)}\\ {\displaystyle c_Q\left(j\right)}& =& {\displaystyle \text{the content of the box containing}\hspace{0.17em}j\hspace{0.17em}\text{in}\hspace{0.17em}{Q}^{\left({\phi }_k\right)},}& \text{(6.12)}\\ {\displaystyle C_Q\left(j\right)}& =& {\displaystyle \frac{q-1}{1-q^{c_Q\left(j\right)-c_Q(j+1)}}}& \text{(6.13)}\end{array}$$

Let $\nu =({\nu }_1,{\nu }_2,\dots ,{\nu }_n)\models n$ with ${\nu }_i\ge 0$ and $\ell \left(\nu \right)=n\text{.}$ Then (a) The irreducible ${𝒯}_{\nu }\text{-modules}$ ${𝒯}_{\nu }^{\lambda }$ are indexed by $\lambda \in {\widehat{𝒯}}_{\nu }\text{.}$ (b) $\text{dim}\left({𝒯}_{\nu }^{\lambda }\right)=\text{Card}\left({\widehat{𝒯}}_{\nu }^{\lambda }\right)\text{.}$ (c) Let ${𝒯}_{\nu }^{\lambda }=ℂ\text{-span}\left\{v_Q\hspace{0.17em}\right|\hspace{0.17em}Q\in {\widehat{𝒯}}_{\nu }^{\lambda }\}\text{.}$ Then $${}^{\nu }{T}_i{v}_Q=q^{-1}{Q}_j(-1)C_Q\left(i\right)v_Q+q^{-1}{Q}_j(-1)(1+C_Q\left(i\right))v_{s_{i}Q},$$ where $v_{s_{i}Q}=0$ if $s_{i}Q$ is not a column strict tableau.

The irreducible modules of ${\mathscr{H}}_1$

The goal of this section is to construct the irreducible ${\mathscr{H}}_1\text{-modules.}$ According to Corollary 6.3 and (6.7), the map $$\begin{array}{cc}{\displaystyle \begin{array}{ccc}\left\{\begin{array}{c}\text{Pairs}\hspace{0.17em}(\nu ,\lambda ),\hspace{0.17em}\nu \models n,\\ {\nu }_i\ge 0,\ell \left(\nu \right)=n,\lambda \in {\widehat{𝒯}}_{\nu }\end{array}\right\}& ⟷& \left\{\begin{array}{c}\text{Irreducible}\\ {\mathscr{H}}_1\text{-modules}\end{array}\right\}\\ (\nu ,\lambda )& \leftrightarrow & {\text{Ind}}_{{𝒯}_{\nu }}^{{\mathscr{H}}_1}\left({𝒯}_{\nu }^{\lambda }\right)\end{array}}& \text{(6.14)}\end{array}$$ is a bijection.

Let $W/\nu$ be the set of minimal length coset representatives of $W/W_{\nu }\text{.}$ Then $$\begin{array}{cc}{\displaystyle \begin{array}{rcl}{\displaystyle {\text{Ind}}_{{𝒯}_{\nu }}^{{\mathscr{H}}_1}\left({𝒯}_{\nu }^{\lambda }\right)}& =& {\displaystyle {\mathscr{H}}_1{\otimes }_{{𝒯}_{\nu }}{𝒯}_{\nu }^{\lambda }}\\ & =& {\displaystyle ℂ\text{-span}\{T_w\otimes v_Q\hspace{0.17em}|\hspace{0.17em}w\in W/\nu \text{, sh}\left(Q\right)=\lambda \text{, wt}\left(Q\right)={\gamma }_{\nu }\}\text{.}}\end{array}}& \text{(6.15)}\end{array}$$ Note that the map $$\begin{array}{cc}{\displaystyle \begin{array}{ccc}\left\{\begin{array}{c}\text{Pairs}\hspace{0.17em}(w,Q),\\ w\in W/\nu ,Q\in {\hat{T}}_{\nu }^{\lambda }\end{array}\right\}& ⟶& \left\{\text{tableau}\hspace{0.17em}Q\hspace{0.17em}\right|\hspace{0.17em}\text{sh}\left(Q\right)=\lambda \text{, wt}\left(Q\right)=\left(1^n\right)\}\\ (w,Q)& ⟼& wQ\end{array}}& \text{(6.16)}\end{array}$$ is a bijection (since $w\in W/\nu$ implies $w$ preserves the relative magnitudes of the entries in $Q^{\left(\phi \right)}$ for all $\phi \text{).}$ For example, under this bijection $$(\begin{array}{c} \end{array},({\begin{array}{c} 1 3 2 4 \end{array}}^{\left({\phi }_1\right)},{\begin{array}{c} 5 6 \end{array}}^{\left({\phi }_2\right)},{\begin{array}{c} 7 \end{array}}^{\left({\phi }_3\right)}))⟷({\begin{array}{c} 2 6 4 7 \end{array}}^{\left({\phi }_1\right)},{\begin{array}{c} 3 5 \end{array}}^{\left({\phi }_2\right)},{\begin{array}{c} 1 \end{array}}^{\left({\phi }_3\right)})$$ (since $2<3$ in $Q^{\left({\phi }_1\right)}$ on the left side, $4<6$ in $Q^{\left({\phi }_1\right)}$ on the right side). Write $$v_{wQ}=T_w\otimes v_Q\text{.}$$ If $\mid \lambda \mid =n,$ then let $$\begin{array}{rcll}{\displaystyle {\hat{\mathscr{H}}}_1^{\lambda }}& =& {\displaystyle \left\{\text{tableau}\hspace{0.17em}Q\hspace{0.17em}\right|\hspace{0.17em}\text{sh}\left(Q\right)=\lambda \text{, wt}\left(Q\right)=\left(1^n\right)\},}& \text{(6.17)}\\ {\displaystyle {\hat{\mathscr{H}}}_1}& =& {\displaystyle \left\{{\widehat{𝔽}}_q^{*}\text{-partitions}\hspace{0.17em}\lambda \hspace{0.17em}\right|\hspace{0.17em}\mid \lambda \mid =n\}\text{.}}& \text{(6.18)}\end{array}$$

Remarks. 1. In the notation of Chapter 5, $${\hat{\mathscr{H}}}_1=\left\{\mathrm{\Theta }\text{-partition}\hspace{0.17em}\lambda \hspace{0.17em}\right|\hspace{0.17em}{\hat{\mathscr{H}}}_1^{\lambda }\ne \varnothing \}\text{.}$$ 2. The map $$\begin{array}{ccc}\left\{\begin{array}{c}\text{Pairs}\hspace{0.17em}(\nu ,\lambda ),\nu \models n,\\ {\nu }_i\ge 0,\ell \left(\nu \right)=n,\lambda \in {\widehat{𝒯}}_{\nu }\end{array}\right\}& ⟷& {\hat{\mathscr{H}}}_1\\ (\nu ,\lambda )& \leftrightarrow & \lambda \end{array}$$ is a bijection, so by (6.14), ${\hat{\mathscr{H}}}_1$ indexes the irreducible ${\mathscr{H}}_1\text{-modules.}$

Suppose $Q$ is a tableau of shape $\lambda$ and weight $\left(1^n\right)\text{.}$ As in (6.11)-(6.13), if $Q^{\left(\phi \right)}$ has the box containing $j,$ then write $$\begin{array}{rcl}{\displaystyle Q_j}& =& {\displaystyle \phi ,}\\ {\displaystyle c_Q\left(j\right)}& =& {\displaystyle \text{the content of the box containing}\hspace{0.17em}j\hspace{0.17em}\text{in}\hspace{0.17em}{Q}^{\left(\phi \right)},}\\ {\displaystyle C_Q\left(j\right)}& =& {\displaystyle \frac{q-1}{1-q^{c_Q\left(j\right)-c_Q(j+1)}}\text{.}}\end{array}$$

(a) The irreducible ${\mathscr{H}}_1\text{-modules}$ ${\mathscr{H}}_1^{\lambda }$ are indexed by $\lambda \in {\hat{\mathscr{H}}}_1\text{.}$ (b) $\text{dim}\left({\mathscr{H}}_1^{\lambda }\right)=\text{Card}\left({\hat{\mathscr{H}}}_1^{\lambda }\right)\text{.}$ (c) Let ${\mathscr{H}}_1^{\lambda }=ℂ\text{-span}\left\{v_Q\hspace{0.17em}\right|\hspace{0.17em}Q\in {\hat{\mathscr{H}}}_1^{\lambda }\}$ as in (6.15) and (6.17). Then $$\begin{array}{rcl}{\displaystyle h{v}_Q}& =& {\displaystyle Q_1\left(h_1\right)Q_2\left(h_2\right)\cdots Q_n\left(h_n\right)v_Q,\text{for}\hspace{0.17em}h=\text{diag}(h_1,h_2,\dots ,h_n)\in T,}\\ {\displaystyle T_i{v}_Q}& =& {\displaystyle \{\begin{array}{ll}{q}^{-1}{v}_{s_{i}Q},& \text{if}\hspace{0.17em}{Q}_i\succ Q_{i+1},\\ Q_i(-1)q^{-1}{C}_Q\left(i\right)v_Q+Q_i(-1)q^{-1}(1+C_Q\left(i\right))v_{s_{i}Q},& \text{if}\hspace{0.17em}{Q}_i=Q_{i+1},\\ v_{s_{i}Q},& \text{if}\hspace{0.17em}{Q}_i\prec Q_{i+1},\end{array}}\end{array}$$ where $v_{s_{i}Q}=0$ if $s_{i}Q$ is not a column strict tableau.

Sample Computations. If $n=10$ for ${\mathscr{H}}_1,$ then $$\begin{array}{rcl}{\displaystyle T_1{v}_{({\begin{array}{c} 3 5 10 4 7 \end{array}}^{\left({\phi }_1\right)},{\begin{array}{c} 1 6 8 \end{array}}^{\left({\phi }_2\right)},{\begin{array}{c} 2 9 \end{array}}^{\left({\phi }_3\right)})}}& =& {\displaystyle v_{({\begin{array}{c} 3 5 10 4 7 \end{array}}^{\left({\phi }_1\right)},{\begin{array}{c} 2 6 8 \end{array}}^{\left({\phi }_2\right)},{\begin{array}{c} 1 9 \end{array}}^{\left({\phi }_3\right)})},}\\ {\displaystyle T_2{v}_{({\begin{array}{c} 3 5 10 4 7 \end{array}}^{\left({\phi }_1\right)},{\begin{array}{c} 1 6 8 \end{array}}^{\left({\phi }_2\right)},{\begin{array}{c} 2 9 \end{array}}^{\left({\phi }_3\right)})}}& =& {\displaystyle q^{-1}{v}_{({\begin{array}{c} 2 5 10 4 7 \end{array}}^{\left({\phi }_1\right)},{\begin{array}{c} 1 6 8 \end{array}}^{\left({\phi }_2\right)},{\begin{array}{c} 3 9 \end{array}}^{\left({\phi }_3\right)})},}\\ {\displaystyle T_3{v}_{({\begin{array}{c} 3 5 10 4 7 \end{array}}^{\left({\phi }_1\right)},{\begin{array}{c} 1 6 8 \end{array}}^{\left({\phi }_2\right)},{\begin{array}{c} 2 9 \end{array}}^{\left({\phi }_3\right)})}}& =& {\displaystyle -{\phi }_1(-1)q^{-1}{v}_{({\begin{array}{c} 3 5 10 4 7 \end{array}}^{\left({\phi }_1\right)},{\begin{array}{c} 1 6 8 \end{array}}^{\left({\phi }_2\right)},{\begin{array}{c} 2 9 \end{array}}^{\left({\phi }_3\right)})}+0,}\\ {\displaystyle T_4{v}_{({\begin{array}{c} 3 5 10 4 7 \end{array}}^{\left({\phi }_1\right)},{\begin{array}{c} 1 6 8 \end{array}}^{\left({\phi }_2\right)},{\begin{array}{c} 2 9 \end{array}}^{\left({\phi }_3\right)})}}& =& {\displaystyle \frac{{\phi }_1(-1)q}{q+1}{v}_{({\begin{array}{c} 3 5 10 4 7 \end{array}}^{\left({\phi }_1\right)},{\begin{array}{c} 1 6 8 \end{array}}^{\left({\phi }_2\right)},{\begin{array}{c} 2 9 \end{array}}^{\left({\phi }_3\right)})}}\\ & & {\displaystyle +{\phi }_1(-1)\left(q^{-1}\frac{q}{q+1}\right)v_{({\begin{array}{c} 3 4 10 5 7 \end{array}}^{\left({\phi }_1\right)},{\begin{array}{c} 1 6 8 \end{array}}^{\left({\phi }_2\right)},{\begin{array}{c} 2 9 \end{array}}^{\left({\phi }_3\right)})}\text{.}}\end{array}$$

A Character Table ${\mathscr{H}}_1$ for $n=2\text{:}$ $$\begin{array}{ccc}{\mathscr{H}}_1& \begin{array}{c} a b \end{array}& \begin{array}{c} a b \end{array}\\ \left({\begin{array}{c} \end{array}}^{\left({\phi }_i\right)}\right)& {\phi }_i\left(ab\right)& {\phi }_i(-ab)\\ \left({\begin{array}{c} \end{array}}^{\left({\phi }_i\right)}\right)& {\phi }_i\left(ab\right)& -{\phi }_i(-ab)q^{-1}\\ ({\begin{array}{c} \end{array}}^{\left({\phi }_i\right)},{\begin{array}{c} \end{array}}^{\left({\phi }_i\right)})& {\phi }_i\left(a\right){\phi }_j\left(b\right)+{\phi }_i\left(b\right){\phi }_j\left(a\right)& 0\end{array}$$

Notes and References

This is an excerpt of the PhD thesis Unipotent Hecke algebras: the structure, representation theory, and combinatorics by F. Nathaniel Edgar Thiem.

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