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

Last updated: 26 March 2015

## The representation theory of the Yokonuma algebra

### General type

Let $1:U\to {ℂ}^{*}$ be the trivial character, and let $e1=1∣U∣ ∑u∈Uu∈ℂG (6.1)$ be the idempotent so that ${\text{Ind}}_{U}^{G}\left(1\right)\cong ℂG{e}_{1}\text{.}$ Then the Yokonuma algebra $ℋ1=EndℂG (ℂGe1)≅ e1ℂGe1$ has a basis ${e1ve1 | v∈N}, indexed byN=⟨ξi,h | i=1,2,…,ℓ, h∈T⟩,$ where ${\xi }_{i}={w}_{i}\left(1\right)={x}_{i}\left(1\right){x}_{-{\alpha }_{i}}\left(-1\right){x}_{i}\left(1\right),$ and $T=⟨{h}_{H}\left(t\right) | H\in {𝔥}_{ℤ}⟩\text{.}$ Recall that ${h}_{i}\left(t\right)={h}_{{H}_{{\alpha }_{i}}}\left(t\right)\text{.}$

The map $ℂT ⟶ ℋ1 h ⟼ e1he1$ is an injective algebra homomorphism (see Chapter 3 (E3)). For $v\in N,$ write $Tv=e1ve1 ∈ℋ1,with Ti=Tξi,and h=Th,h∈T.$

(Yokonuma). ${ℋ}_{1}$ is generated by ${T}_{i},$ $i=1,\dots ,\ell ,$ and $h\in T$ with relations $Tih = si(h)Ti, Ti2 = q-1hi(-1) +q-1∑t∈𝔽q* hi(t)Ti, TiTjTi⋯ ⏟mij = TjTiTj⋯ ⏟mij , where mij is the order of sisj in W, ThTk = Thk,h,k∈T.$  Proof.

Consider the product ${\xi }_{i}{\xi }_{j}\in N$ for $i\ne j\text{.}$ Note that $e1ξiξje1 = e1(∏α≠αieα) ξiξjeαie1 (by (3.17)) = e1ξi (∏α≠αieα) eαieξje1 (by Chapter 3, (E1)) = e1ξie1ξje1 = (e1ξie1) (e1ξje1).$ Thus, if $v={v}_{1}{v}_{2}\cdots {v}_{r}{v}_{T}\in N$ decomposes according to ${s}_{{i}_{1}}{s}_{{i}_{2}}\cdots {s}_{{i}_{r}}\in W$ (see (3.4)), then $Tv=e1v1 v2⋯vrvTe1 =Ti1Ti2⋯ TirvT. (6.2)$ The Yokonuma algebra is therefore generated by ${T}_{i},$ $i=1,2,\dots ,\ell ,$ and $h\in T\text{.}$

The necessity of the relations is a direct consequence of (6.2), (N3), (UN3), and (N2) (the last three are from Chapter 3). For sufficiency, use a similar argument to the one used in the proof of Theorem 3.5 in [IMa1965].

$\square$

#### A reduction theorem

Let $\stackrel{ˆ}{T}$ index the irreducible $ℂT\text{-modules.}$ Since $T$ is abelian, all the irreducible modules are one-dimensional, and we may identify the label $\gamma \in \stackrel{ˆ}{T}$ of ${V}^{\gamma }=ℂ\text{-span}\left\{{v}_{\gamma }\right\}$ with the homomorphism $\gamma :T\to {ℂ}^{*}$ given by $hvγ=γ(h) vγ,for h∈T.$ Suppose $V$ is an ${ℋ}_{1}\text{-module.}$ As a $T\text{-module}$ $V=⨁γ∈Tˆ Vγ,where Vγ ={v∈V | hv=γ(h)v, h∈T}.$

Note that $ℋ1=⨁γ∈Tˆ ℋ1τγ,where τγ=1∣T∣ ∑h∈Tγ(h-1)h. (6.3)$ Recall the surjection $\pi :N\to W$ of (3.3). Use this map to identify $w=si1si2⋯ sir∈W⟷ w=ξi1ξi2⋯ ξir∈N,for r minimal. (6.4)$ Thus, the $W\text{-action}$ on $\stackrel{ˆ}{T}$ $(wγ)(h)= γ(w-1(h)), for w∈W, h∈T , and γ∈Tˆ,$ implies that for $\gamma \in \stackrel{ˆ}{T},$ ${v}_{\gamma }\in {V}_{\gamma },$ $hTwvγ=Tww-1 (h)vγ=γ (w-1h)Twvγ for all h∈T.$ Thus, $Tw(Vγ)⊆ Vwγ. (6.5)$ Let $Wγ= {w∈W | w(γ)=γ}$ and $𝒯γ=τγ ℋ1τγ=ℂ -span{τγTwτγ | w∈Wγ}. (6.6)$

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

Let $V$ be an irreducible ${ℋ}_{1}\text{-module}$ 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 ${ℋ}_{1}{\otimes }_{{𝒯}_{\gamma }}M$ is an irreducible ${ℋ}_{1}\text{-module;}$ (c) the map $Ind𝒯γℋ1(Vγ) = ℋ1⊗𝒯γVγ ⟶∼ V Tw⊗vγ ⟼ Twvγ$ is an ${ℋ}_{1}\text{-module}$ isomorphism.  Proof.

(a) Let $0\ne {v}_{\gamma }\in {V}_{\gamma }\text{.}$ The irreducibility of $V$ implies that ${ℋ}_{1}{v}_{\gamma }=V,$ so for any $v\in {V}_{\gamma },$ there exist elements ${c}_{w,\nu }\in ℂ$ such that $v = ∑w∈Wν∈Tˆ cw,νTwτνvγ (by (6.3)) = ∑w∈Wcw,γ Twτγvγ (by the orthogonality of characters of T) = ∑w∈Wγcw,γ τγTwτγvγ. (by (6.5) and since v∈Vγ)$ Since an arbitrarily chosen $v$ is contained in ${𝒯}_{\gamma }{v}_{\gamma },$ we have ${𝒯}_{\gamma }{v}_{\gamma }=V\gamma ,$ making ${V}_{\gamma }$ an irreducible ${𝒯}_{\gamma }\text{-module.}$

(b) Suppose $M$ is an irreducible ${𝒯}_{\gamma }\text{-module.}$ Let $V = Ind𝒯γℋ1(M) = ℋ1⊗𝒯γM = ℂ-span { Twτν⊗v | w∈W,ν≠γ∈Tˆ,v∈M } = ℂ-span { Tw⊗v | w∈W/ Wγ,v∈M }$ where the last equality follows from $τν⊗v=τν⊗ τγv=τντγ ⊗v=0⊗v,for ν≠γ.$ Note that ${V}_{\gamma }=ℂ\text{-span}\left\{{e}_{1}\otimes v | v\in M\right\}\cong M\text{.}$ In particular, ${V}_{\gamma }$ is an irreducible ${𝒯}_{\gamma }\text{-module}$ and ${ℋ}_{1}{V}_{\gamma }=V\text{.}$

Suppose $V$ has a nontrivial ${ℋ}_{1}\text{-submodule}$ $V\prime \text{.}$ Since $V$ is induced from ${V}_{\gamma },$ there must by some ${T}_{w}\in {ℋ}_{1}$ such that $Tw(V′)∩ Vγ≠0.$ But $V\prime$ is an ${ℋ}_{1}\text{-module,}$ so $V\prime \cap {V}_{\gamma }\ne 0\text{.}$ As an irreducible ${𝒯}_{\gamma }\text{-module}$ ${V}_{\gamma }\subseteq V\prime ,$ and by the construction of $V,$ $ℋ1V′⊇ ℋ1Vγ=V.$ Therefore, $V$ contains no nontrivial, proper submodules, making $V$ an irreducible ${ℋ}_{1}\text{-module.}$

(c) Follows from the proof of (b).

$\square$

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

The map ${Irreducible ℋ1-modules} ⟷ {Pairs (γ,λ),γ∈Tˆ/W,λ∈𝒯ˆγ} Ind𝒯γℋ1(𝒯γλ) ↔ (γ,λ)$ is a bijection.

#### The algebras ${𝒯}_{\gamma }$

Let $B=UT$ be a Borel subgroup of $G\text{.}$ Since $B ⟶ T uh ⟼ h$ 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 $e1τγ=1∣B∣ ∑b∈Bγ(b-1)b (e1 as in (6.1)).$ Thus, $𝒯γ=τγe1 ℂGe1τγ≅ EndℂG(IndBG(γ)) =ℋ(G,B,γ),$ where if $\gamma$ is the trivial character ${1}_{B},$ then $ℋ\left(G,B,{1}_{B}\right)$ is the Iwahori-Hecke algebra.

Let $\gamma \in \stackrel{ˆ}{T}$ be such that ${W}_{\gamma }$ is generated by simple reflections. Then ${𝒯}_{\gamma }$ is presented by generators $\left\{{T}_{i} | {s}_{i}\in {W}_{\gamma }\right\}$ with relations $Ti2 = { q-1+q-1 (q-1)Ti, if γ(hαi(t)) =1 for all t∈𝔽q*, γ(hαi(-1)) q-1, otherwise, TiTjTi⋯ ⏟mij = TjTiTj⋯ ⏟mij ,where mij is the order of sisj in W.$  Proof.

This follows from Theorem 6.1 and $(∑t∈𝔽q*hαi(t)) τγ=∑t∈𝔽q* γ(hαi(t))τγ= { (q-1)τγ, if γ(hαi(t))= 1 for all t∈𝔽q*, 0, otherwise.$

$\square$

### 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 $G={GL}_{n}\left({𝔽}_{q}\right),$ ${ℋ}_{1}$ has generators ${T}_{i},$ ${h}_{{\epsilon }_{j}}\left(t\right),$ for $1\le i $1\le j\le n$ and $t\in {𝔽}_{q}^{*}$ with relations $Ti2 = q-1+q-1 hεi(-1) ∑t∈𝔽q* hεi(t) hεi+1 (t-1)Ti, TiTi+1Ti = Ti+1TiTi+1, TiTj=TjTi, ∣i-j∣>1, hεj(t)Ti = Tihεj′(t), where j′=si(j), hεi(a) hεi(b) = hεi(ab), hεi(a) hεj(b)= hεj(b) hεi(a), a,b∈𝔽q*.$

The Yokonuma algebra has a decomposition $ℋ1=⨁γ∈Tˆ ℋ1τγ,where τγ=1∣T∣ ∑h∈Tγ(h-1)h.$

Fix a total ordering $\le$ on the set ${\stackrel{ˆ}{𝔽}}_{q}^{*}=\left\{{\phi }_{1},{\phi }_{2},\dots ,{\phi }_{q-1}\right\}$ of linear characters of ${𝔽}_{q}^{*},$ such that ${\phi }_{1}:{𝔽}_{q}^{*}\to {ℂ}^{*}$ is the trivial character. Suppose $\gamma \in \stackrel{ˆ}{T}\text{.}$ Since $⟨{h}_{{\epsilon }_{i}}\left(t\right) | t\in {𝔽}_{q}^{*}⟩\cong {𝔽}_{q}^{*},$ $γ(h) = γ(hε1(h1)) γ(hε2(h2))⋯ γ(hεn(hn)), where h=diag(h1,h2,…,hn)∈T, = γ1(h1) γ2(h2)⋯ γn(hn), where γi∈𝔽ˆq*.$ Thus, every $\gamma \in \stackrel{ˆ}{T}$ can be written $\gamma ={\phi }_{{i}_{1}}\otimes {\phi }_{{i}_{2}}\otimes \cdots \otimes {\phi }_{{i}_{n}}\text{.}$

Let $\nu =\left({\nu }_{1},{\nu }_{2},\dots ,{\nu }_{n}\right)\models n$ with ${\nu }_{i}\ge 0\text{.}$ Write $γν= φ1⊗⋯⊗φ1 ⏟ν1 terms ⊗ φ2⊗⋯⊗φ2 ⏟ν2 terms ⊗⋯⊗ φn⊗⋯⊗φn ⏟νn terms andτν= τγν. (6.7)$ Note that the ${\gamma }_{\nu }$ are orbit representatives of the $W\text{-action}$ in $\stackrel{ˆ}{T}\text{.}$ Let $Wν = {w∈W | w(γν)=γν}= Wν1⊕Wν2⊕⋯⊕Wνn, 𝒯ν = τνℋ1τν= ℂ-span{τνTwτν | w∈Wν}.$ 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 $si∈Wνk.$

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 ${ νTi | si∈Wν }$ with relations $νTi2 = q-1+φk (-1)q-1 (q-1)νTi, for si∈Wνk νTiνTi+1 νTi = νTi+1ν TiνTi+1 νTiνTj = νTjνTi, for ∣i-j∣>1.$  Proof.

Note that if ${s}_{i}\in {W}_{\nu }$ then ${\gamma }_{\nu }\left({h}_{{\epsilon }_{i}}\left(t\right)\right)={\gamma }_{\nu }\left({h}_{{\epsilon }_{i+1}}\left(t\right)\right)\text{.}$ Thus, the lemma follows from the Yokonuma algebra relations.

$\square$

The Iwahori-Hecke algebra $𝒯\subseteq {ℋ}_{1}$ is the algebra $𝒯n=ℋ(n), (recall φ1 is the trivial character).$ In ${𝒯}_{n},$ write $Ii=τ(n)Ti τ(n).$

Let $\nu =\left({\nu }_{1},\dots ,{\nu }_{n}\right)\models n\text{.}$ Index the generators ${I}_{i}$ of ${𝒯}_{{\nu }_{k}}$ by $\left\{i | {s}_{i}\in {W}_{{\nu }_{k}}\right\}\text{.}$ Then the map $𝒯ν ⟶∼ 𝒯ν1⊗𝒯ν2⊗⋯⊗𝒯νn νTi ⟼ φk(-1)1⊗⋯⊗1⏟k-1 terms⊗Ii⊗1⊗⋯⊗1⏟n-k terms,for si∈Wνk,$ is an algebra isomorphism.  Proof.

Let $ν(k)= ( 0,…,0⏟k-1 terms, νk, 0,…,0⏟n-k terms ) ⊨νk.$ Then the map $𝒯ν(k) ⟶ 𝒯νk ν(k)Ti ⟼ φk(-1)Ii$ is an algebra isomorphism by Lemma 6.5. Corollary 6.6 follows by applying this map to tensor products.

$\square$

#### 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 $cP(i) = the content of the box containing i, CP(i) = q-11-qcP(i)-cP(i+1). (6.8)$

([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) $\text{dim}\left({𝒯}_{n}^{\mu }\right)=\text{Card}\left\{\text{standard tableau} P | \text{sh}\left(P\right)=\mu \right\},$ (c) Let ${𝒯}_{n}^{\mu }=ℂ\text{-span}\left\{{v}_{P} | \text{sh}\left(P\right)=\mu \text{, wt}\left(P\right)=\left({1}^{n}\right)\right\}\text{.}$ Then $IivP=q-1CP (i)vP+q-1 (1+CP(i)) vsiP$ where ${v}_{{s}_{i}P}=0$ if ${s}_{i}P$ is not a column strict tableau.

Recall from Chapter 5, an ${\stackrel{ˆ}{𝔽}}_{q}^{*}\text{-partition}$ $\lambda =\left({\lambda }^{\left({\phi }_{1}\right)},{\lambda }^{\left({\phi }_{2}\right)},\dots ,{\lambda }^{\left({\phi }_{q-1}\right)}\right)$ is a sequence of partitions indexed by $\stackrel{ˆ}{𝔽}q*\text{.}$ Let $∣λ∣= ∣λ(φ1)∣+ ∣λ(φ2)∣+⋯+ ∣λ(φq-1)∣= total number of boxes.$

Let $\nu =\left({\nu }_{1},{\nu }_{2},\dots ,{\nu }_{n}\right)\models n$ with ${\nu }_{i}\ge 0$ ${\gamma }_{\nu }$ as in (6.7). A standard $\nu \text{-tableau}$ $Q= ( Q(φ1), Q(φ2),…, Q(φq-1) )$ 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 $𝒯ˆνλ = {standard ν-tableau Q | sh(Q)=λ}, (6.9) 𝒯ˆν = {𝔽ˆq*-partition λ | 𝒯ˆνλ≠∅} = {𝔽ˆq*-partition λ | ∣λ(φi)∣=∣νi∣}. (6.10)$

Example. If $\nu =\left(3,0,1,0\right),$ then ${\gamma }_{\nu }={\phi }_{1}\otimes {\phi }_{1}\otimes {\phi }_{1}\otimes {\phi }_{3}\text{.}$ The set ${\stackrel{ˆ}{𝒯}}_{\nu }$ is ${ ( (φ1) , ∅(φ2), (φ3) , ∅(φ4) ) , ( (φ1) , ∅(φ2), (φ3) , ∅(φ4) ) , ( (φ1) , ∅(φ2), (φ3) , ∅(φ4) ) } ,$ and, for example, $𝒯ˆ ν ( (φ1) , (φ3) ) = { ( 1 2 3 (φ1) , ∅(φ2), 4 (φ3) , ∅(φ4) ) , ( 1 3 2 (φ1) , ∅(φ2), 4 (φ3) , ∅(φ4) ) }$

Let $Q\in {\stackrel{ˆ}{𝒯}}_{\nu }^{\lambda }\text{.}$ Suppose ${s}_{j}\in {W}_{{\nu }_{k}}$ so that $j\in {Q}^{\left({\phi }_{k}\right)}\text{.}$ Write $Qj = φk, (6.11) cQ(j) = the content of the box containing j in Q(φk), (6.12) CQ(j) = q-11-qcQ(j)-cQ(j+1) (6.13)$

Let $\nu =\left({\nu }_{1},{\nu }_{2},\dots ,{\nu }_{n}\right)\models n$ 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 $\lambda \in {\stackrel{ˆ}{𝒯}}_{\nu }\text{.}$ (b) $\text{dim}\left({𝒯}_{\nu }^{\lambda }\right)=\text{Card}\left({\stackrel{ˆ}{𝒯}}_{\nu }^{\lambda }\right)\text{.}$ (c) Let ${𝒯}_{\nu }^{\lambda }=ℂ\text{-span}\left\{{v}_{Q} | Q\in {\stackrel{ˆ}{𝒯}}_{\nu }^{\lambda }\right\}\text{.}$ Then $νTivQ=q-1 Qj(-1)CQ(i) vQ+q-1Qj (-1)(1+CQ(i)) vsiQ,$ where ${v}_{{s}_{i}Q}=0$ if ${s}_{i}Q$ is not a column strict tableau.  Proof.

Transfer the explicit action of Theorem 6.7 across the isomorphism ${𝒯}_{\nu }\cong {𝒯}_{{\nu }_{1}}\otimes {𝒯}_{{\nu }_{2}}\otimes \cdots \otimes {𝒯}_{{\nu }_{n}}$ of Corollary 6.6.

$\square$

#### The irreducible modules of ${ℋ}_{1}$

The goal of this section is to construct the irreducible ${ℋ}_{1}\text{-modules.}$ According to Corollary 6.3 and (6.7), the map ${ Pairs (ν,λ), ν⊨n, νi≥0,ℓ(ν)=n,λ∈𝒯ˆν } ⟷ {Irreducibleℋ1-modules} (ν,λ) ↔ Ind𝒯νℋ1(𝒯νλ) (6.14)$ is a bijection.

Let $W/\nu$ be the set of minimal length coset representatives of $W/{W}_{\nu }\text{.}$ Then $Ind𝒯νℋ1 (𝒯νλ) = ℋ1⊗𝒯ν 𝒯νλ = ℂ-span { Tw⊗vQ | w∈W/ν, sh (Q)=λ, wt (Q)=γν } . (6.15)$ Note that the map ${Pairs (w,Q),w∈W/ν,Q∈Tˆνλ} ⟶ {tableau Q | sh(Q)=λ, wt(Q)=(1n)} (w,Q) ⟼ wQ (6.16)$ 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 $( , ( 1 3 2 4 (φ1) , 5 6 (φ2) , 7 (φ3) ) ) ⟷ ( 2 6 4 7 (φ1) , 3 5 (φ2) , 1 (φ3) )$ (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 $vwQ=Tw⊗vQ.$ If $\mid \lambda \mid =n,$ then let $ℋˆ1λ = {tableau Q | sh(Q)=λ, wt(Q)=(1n)}, (6.17) ℋˆ1 = { 𝔽ˆq* -partitions λ | ∣λ∣=n } . (6.18)$

Remarks. 1. In the notation of Chapter 5, $ℋˆ1= {Θ-partition λ | ℋˆ1λ≠∅}.$ 2. The map ${Pairs (ν,λ),ν⊨n,νi≥0,ℓ(ν)=n,λ∈𝒯ˆν} ⟷ ℋˆ1 (ν,λ) ↔ λ$ is a bijection, so by (6.14), ${\stackrel{ˆ}{ℋ}}_{1}$ indexes the irreducible ${ℋ}_{1}\text{-modules.}$

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 $Qj = φ, cQ(j) = the content of the box containing j in Q(φ), CQ(j) = q-11-qcQ(j)-cQ(j+1).$

(a) The irreducible ${ℋ}_{1}\text{-modules}$ ${ℋ}_{1}^{\lambda }$ are indexed by $\lambda \in {\stackrel{ˆ}{ℋ}}_{1}\text{.}$ (b) $\text{dim}\left({ℋ}_{1}^{\lambda }\right)=\text{Card}\left({\stackrel{ˆ}{ℋ}}_{1}^{\lambda }\right)\text{.}$ (c) Let ${ℋ}_{1}^{\lambda }=ℂ\text{-span}\left\{{v}_{Q} | Q\in {\stackrel{ˆ}{ℋ}}_{1}^{\lambda }\right\}$ as in (6.15) and (6.17). Then $hvQ = Q1(h1) Q2(h2)⋯ Qn(hn)vQ, for h=diag (h1,h2,…,hn) ∈T, TivQ = { q-1vsiQ, if Qi≻ Qi+1, Qi(-1)q-1 CQ(i)vQ+ Qi(-1)q-1 (1+CQ(i)) vsiQ, if Qi= Qi+1, vsiQ, if Qi ≺Qi+1,$ where ${v}_{{s}_{i}Q}=0$ if ${s}_{i}Q$ is not a column strict tableau.  Proof.

(a) follows from Remark 2, and (b) follows from (6.15) and (6.17).)

(c) Directly compute the action of ${ℋ}_{1}$ on $ℋ1λ= Ind𝒯νℋ1 (𝒯νλ)= ℂ-span {Tw⊗vQ | w∈W/λ∼,Q∈𝒯ˆνλ}$ For $1\le i if $\ell \left({s}_{i}w\right)<\ell \left(w\right),$ then $TiTw⊗vQ = Ti2Tsiw⊗ vQ = q-1Tsiw⊗ vQ+q-1hεi (-1)∑t∈𝔽q* hεi(t) hεi+1 (t-1)Tw⊗vQ = q-1Tsiw⊗vQ +q-1(wQ)i(-1) ∑t∈𝔽q* (wQ)i(t) (wQ)i+1(t-1) Tw⊗vQ$ Since $\ell \left({s}_{i}w\right)<\ell \left(w\right)$ can hold only if ${\left(wQ\right)}_{i}\ne {\left(wQ\right)}_{i+1},$ the orthogonality of characters implies $TiTw⊗vQ=q-1 Tsiw⊗vQ+0. (6.19)$ If $\ell \left({s}_{i}w\right)>\ell \left(w\right),$ then there are two cases:

Case 1: $\ell \left({s}_{i}w{s}_{j}\right)<\ell \left({s}_{i}w\right)$ for some ${s}_{j}\in {W}_{\stackrel{\sim }{\gamma }},$

Case 2: $\ell \left({s}_{i}w{s}_{j}\right)>\ell \left({s}_{i}w\right)$ for all ${s}_{j}\in {W}_{\stackrel{\sim }{\gamma }}\text{.}$

In Case 1, $TiTw⊗vQ = Tsiw⊗vQ = TsiwsjTj ⊗vQ = Tsiwsj⊗ TjvQ = Tsiwsj⊗ Qj(-1)q-1 CQ(j)vQ+ Qj(-1)q-1 (1+CQ(j)) vsjQ = Qj(-1)q-1 CQ(j)Tsiwsj ⊗vQ+Qj(-1) q-1(1+CQ(j)) Tsiwsj⊗vsjQ. (6.20)$ In Case 2, $TiTw⊗vQ= Tsiw⊗vQ. (6.21)$ Make the identification of (6.16) in equations (6.19), (6.20), and (6.21) to obtain the ${ℋ}_{1}\text{-action}$ on ${ℋ}_{1}^{\lambda }\text{.}$

$\square$

Sample Computations. If $n=10$ for ${ℋ}_{1},$ then $T1 v ( 3 5 10 4 7 (φ1) , 1 6 8 (φ2) , 2 9 (φ3) ) = v ( 3 5 10 4 7 (φ1) , 2 6 8 (φ2) , 1 9 (φ3) ) , T2 v ( 3 5 10 4 7 (φ1) , 1 6 8 (φ2) , 2 9 (φ3) ) = q-1 v ( 2 5 10 4 7 (φ1) , 1 6 8 (φ2) , 3 9 (φ3) ) , T3 v ( 3 5 10 4 7 (φ1) , 1 6 8 (φ2) , 2 9 (φ3) ) = -φ1(-1)q-1 v ( 3 5 10 4 7 (φ1) , 1 6 8 (φ2) , 2 9 (φ3) ) +0 , T4 v ( 3 5 10 4 7 (φ1) , 1 6 8 (φ2) , 2 9 (φ3) ) = φ1(-1)q q+1 v ( 3 5 10 4 7 (φ1) , 1 6 8 (φ2) , 2 9 (φ3) ) +φ1(-1) (q-1qq+1) v ( 3 4 10 5 7 (φ1) , 1 6 8 (φ2) , 2 9 (φ3) ) .$

A Character Table ${ℋ}_{1}$ for $n=2\text{:}$ $ℋ1 a b a b ( (φi) ) φi(ab) φi(-ab) ( (φi) ) φi(ab) -φi(-ab)q-1 ( (φi) , (φi) ) φi(a)φj(b)+φi(b)φj(a) 0$

## 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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