The 0 Hecke algebra

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

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 10 June 2010

The 0 Hecke algebra

Let $W$ be a Weyl group with simple reflections $s_{1}, \dots, s_{n} .$ The $0$ -Hecke algebra is given by generators $T_{1}, \dots, T_{n}$ and relations $T_{i}^{2} = - T_{i} and T_{i} T_{j} T_{i} \dots (m_{i j} factors) = T_{j} T_{i} T_{j} \dots (m_{i j} factors), for i \neq j,$ where $m_{i j}$ is the order of $s_{i} s_{j}$ in $W .$ If $e_{i} = - T_{i}, and f_{i} = 1 - e_{i} = 1 + T_{i},$ then $e_{i}^{2} = e_{i}, f_{i}^{2} = f_{i}, e_{i} e_{j} e_{i} \dots (m_{i j} factors) = e_{j} e_{i} e_{j} \dots (m_{i j} factors), f_{i} f_{j} f_{i} \dots (m_{i j} factors) = f_{j} f_{i} f_{j} \dots (m_{i j} factors) .$ The last identity is proved by noting that a term of the form $1.1 (k factors) \dots . e_{i} . 1.1 \dots (B factors)$ in the product $f_{i} f_{j} \dots = (1 - e_{i}) (1 - e_{j}) \dots$ cancels with the term $1 \dots 1 . e_{i} . 1 . e_{i} \dots (B factors) .$ The remaining terms are products of the form ${(- 1)}^{m_{i j} - k} (1 \dots 1 (k factors)) (e_{i} e_{j} e_{i} \dots (m_{i j} - k factors)) and {(- 1)}^{m_{i j} - k - 1} (1 \dots 1 (k factors)) (e_{i} e_{j} e_{i} \dots (m_{i j} - k - 1 factors)) . 1 .$ Thus $\begin{array}{rcl} f_{i} f_{j} \dots & = & 1 - e_{i} - e_{j} + e_{i} e_{j} + e_{j} e_{i} - e_{i} e_{j} e_{i} - e_{j} e_{i} e_{j} + \dots + (e_{i} e_{j} \dots (m_{i j} factors)) \\ = & 1 + (\sum_{k = 1}^{m_{i j} - 1} (e_{i} e_{j} \dots (k factors)) {(- 1)}^{k}) + e_{i} e_{j} \dots (m_{i j} factors) \\ = & 1 + (\sum_{k = 1}^{m_{i j} - 1} (e_{i} e_{j} \dots (k factors)) {(- 1)}^{k}) + e_{j} e_{i} \dots (m_{i j} factors) \\ = & f_{j} f_{i} \dots . \end{array}$

Irreducible representations

Let $V$ be a simple $H (0)$ module and let $v \in V, v \neq 0 .$ Then $ℂ e_{w_{0}} v is a submodule of V .$ So $V = ℂ e_{w_{0}} v$ or $e_{w_{0}} v = 0 .$ If $e_{w_{0}} v = 0,$ $ℂ e_{s_{i} w_{0}} v$ is a submodule of $V, 1 \leq i \leq n .$ So $V = ℂ e_{s_{i} w_{0}} v$ for some $i,$ or all $e_{s_{i} w_{0}} v = 0 .$ Thus, by descending induction on $l (w)$ we find $V = ℂ e_{w} v,$ for some $w \in W .$ So $V$ is one dimensional.

Let $J \subseteq \{1 \dots n\}$ and define $χ^{J} : H (0) \to ℂ by χ^{J} (e_{i}) = \{\begin{array}{rcl} 1, & if & i \in J, \\ 0, & if & i \notin J . \end{array}$ This defines $2^{n}$ irreducible one dimensional representations of $H (0) .$ By the argument in the first paragraph, all irreducible representations of $H (0)$ are one dimensional. The relation $e_{i}^{2} = e_{i}$ forces $χ (e_{i}) = 0$ or $χ (e_{i}) = 1$ for al one dimensional representation $χ : H (0) \to ℂ,$ so that $χ^{J}, J \subseteq \{12 \dots n\}$ are a complete set of irreducible $H (0)$ representations.

The radical of $H (0)$

For each $i, j, 1 \leq i, j \leq n, i \neq j,$ and each $J \subseteq \{12 \dots n\},$ $χ^{J} (e_{i} e_{j} - e_{i} e_{j} e_{i}) = 0 .$ So the element $e_{i} e_{j} - e_{i} e_{j} e_{i}$ acts by 0 on every irreducible $H (0)$ -module. So $e_{i} e_{j} - e_{i} e_{j} e_{i} \in Rad (H (0)) .$ If H = H 0 ei ej - ei ej ei and ei is the image of $e_{i}$ in H then $\begin{array}{rcl} \end{array}$ ei ej = ei ej ei = ei ej ei ej =…= ei ej ei … m ij factors = ej ei ej … m ij factors =…= ej ei , for $i \neq j .$ So H is a commutative algebra. In view of the relation ei 2 = ei the elements e i1 … e ik ,1≤ i1 <…≤ ik ≤n,span H and so $\dim ()$ H < 2n . Since all $2^{n}$ irreducible representations $χ^{J}$ of $H (0)$ are representations of H , $\dim ()$ H = 2n and H is semisimple. So $Rad (H (0)) = ⟨e_{i} e_{j} - e_{i} e_{j} e_{i} | i \neq j⟩ .$

Projective indecomposable $H (0)$ modules

For $w \in W$ let $e_{w} = e_{i_{1}} \dots e_{i_{p}}$ and $f_{w} = f_{i_{1}} \dots f_{i_{p}}$ for a reduced decomposition $w = s_{i_{1}} \dots s_{i_{p}}$ of $w .$

Let $J \subseteq \{12 \dots n\}$ and define $P (J) = H (0) e_{w_{j}} f_{w_{J^{c}}},$ where $w_{J}$ and $w_{J^{c}}$ are the longest elements of the parabolic subgroups $W_{J}$ and $W_{J^{c}},$ respectively. The $P (J)$ has basis $\{e_{w} f_{w_{J^{c}}} | D_{r} (w) = J\}, where D_{r} (w) = \{i | w s_{i} < w\} .$ The Cartan invariants are $\begin{array}{rcl} c_{J K} & = & (mutliplicity of L (K) in a composition series of P (J)) \\ = & Card \{w \in W | D_{r} (w) = J, D_{l} (w) = K\}, \end{array}$ where $D_{l} (w) = \{i | s_{i} w < w\} .$ Since $e_{w} f_{w_{J^{c}}} = e_{w} + \sum_{v > w} c_{v} e_{v}, for some c_{v} \in ℂ,$ it follows that $H (0) = \underset{J \subseteq \{12 \dots n\}}{\oplus} P (J), as H (0) - modules.$ Since $P (J)$ has head isomorphic to $L (J) ≅ \frac{P (J)}{span \{e_{w} e_{w_{J}} f_{w_{J^{c}}} | w > 1\}},$ the $P (J)$ are the projective indecomposable modules (PIMs) of $H (0) .$ The matrix $C = (c_{J K}) is the Cartan matrix for H (0) .$

Decomposition numbers

Let $L_{q} (λ)$ be the irreducible $H (q)$ modules in a form which can be specialised at $q = 0 .$ Let $L_{0} (K)$ denote the irreducible $H (0)$ module indexed by $K .$ The decomposition numbers are $\begin{array}{rcl} d_{λ K} & = & (multiplicity of L (K) in a composition series of Δ_{0} (λ)) \\ = & Card \{w \in ℱ^{λ} | D_{l} (w) = K\}, \end{array}$ and $c_{J K} = \sum_{λ} d_{λ J} d_{λ K}, or, in matrix notation, C = D^{t} D,$ where $C = (c_{J K}) is the Cartan matrix, and$ $D = (d_{λ K}), is the decomposition matrix.$

This can be checked directly when $H$ is the dihedral Hecke algebra by noting that the irreducible representations of $H$ given in (???_ specialise at $q = 0 .$ They are given explicitly by $ρ (T_{1}) = (\begin{matrix} \frac{- (1 + ξ^{k})}{ξ^{k} - ξ^{- k}} & \frac{- (1 + ξ^{k}) (1 + ξ^{k})}{{(ξ^{k} - ξ^{- k})}^{2}} \\ ξ^{- k} & \frac{1 + ξ^{k}}{ξ^{k} - ξ^{- k}} \end{matrix}),$ $ρ (T_{2}) = (\begin{matrix} \frac{- (1 + ξ^{k})}{ξ^{k} - ξ^{- k}} & \frac{- (1 + ξ^{- k}) (1 + ξ^{- k})}{{(ξ^{k} - ξ^{- k})}^{2}} \\ ξ^{- k} & \frac{1 + ξ^{k}}{ξ^{k} - ξ^{- k}} \end{matrix}) .$ Then, in this representation, $- T_{2} projects onto v_{2} = (\begin{matrix} \frac{1 + ξ^{k}}{ξ^{k} - ξ^{- k}} \\ 1 \end{matrix}),$ and $T_{1} v_{2} = 0 .$ Similarly, $- T_{2} projects onto v_{2} = (\begin{matrix} \frac{1 + ξ^{k}}{ξ^{k} - ξ^{- k}} \\ - ξ^{- k} \end{matrix}),$ and $T_{2} v_{1} = \frac{(1 - x i^{- 2 k}) (1 + ξ^{k})}{ξ^{k} - ξ^{- k}} (\begin{matrix} \frac{1 + ξ^{k}}{ξ^{k} - ξ^{- k}} \\ 1 \end{matrix}),$ which is a multiple of $v_{2}$ (equal to 0 mod $v_{2}$ ). From this one decudes the decomposition matrices as in (???).

Cell modules

Let $Δ (λ)$ be the cell module of $H (q)$ indexed by the cell $λ$ and let $Δ_{0} (λ)$ be its specialisation at $q = 0 .$ The module $Δ_{0} (λ)$ has basis $\{\}$ Cw | w∈ ℱ λ where ℱ λ is the left cell in W indexed by λ. By [KL, 2.3 (a)-2.3(c)], $T_{s_{i}} C_{w} = \{\begin{array}{rcl} - C_{w} & if & s_{i} w < w, \\ q C_{w} + q^{\frac{1}{2}} C_{s_{i} w} + q^{\frac{1}{2}} \sum_{s_{i} z < z} μ (z, w) C_{z}, & if & s_{i} w > w, \end{array}\}$ where $μ (z, w)$ is the coefficient of $q^{\frac{1}{2} (l (w) - l (z) - 1)}$ in the Kazhdan-Lusztig polynomial $P_{z, w} .$ The cell decomposition numbers $\begin{array}{rcl} κ_{λ K} & = & (multiplicity of L_{0} (K) in a decomposition series of Δ_{0} (λ)) \\ = & Card \{w \in ℱ^{λ} | D_{l} (w) = K\} . \end{array}$ The decomposition of $W$ into left and right cells $\begin{array}{rcl} W & \tilde{\leftrightarrow} & (left cell, right cell) \\ w & \mapsto & (P (w), Q (w)) \end{array}$ gives the formula????? $c_{J K} = \sum_{λ} κ_{λ J} κ_{λ K}, (C = κ^{t} κ),$ where $C = (c_{J K}), is the Cartan matrix, and$ $κ = (κ_{λ K}) is the cell decomposition matrix.$

For dihedral groups, $I_{2} (m),$ the left cells are $Γ^{\emptyset} = \{1\}, Γ^{1} = \{s_{1} s_{2} s_{1} s_{1} s_{2} s_{1} s_{2} s_{1} s_{2} s_{1} \dots \dots s_{1} s_{2} s_{1} (m factors)\},$ $Γ^{2} = \{s_{2} s_{1} s_{2} s_{2} s_{1} s_{2} s_{1} s_{2} s_{1} s_{2} \dots \dots s_{2} s_{1} s_{2} (m factors)\}, Γ^{12} = \{w_{0}\}$ and at $q = 0$ the cell representations have matrices $Δ_{0}^{\emptyset} (T_{1}) = (- 1),$ $Δ_{0}^{\emptyset} (T_{2}) = (- 1),$ $Δ_{0}^{1} (T_{1}) = (\begin{matrix} 0 \\ - 1 \\ 0 \\ - 1 \\ ⋱ \end{matrix}),$ $Δ_{0}^{1} (T_{2}) = (\begin{matrix} - 1 \\ 0 \\ - 1 \\ 0 \\ ⋱ \end{matrix}),$ are $(m - 1) \times (m - 1)$ matrices with rows and columns indexed by $C_{1}, C_{21}, C_{121}, \dots,$ $Δ_{0}^{2} (T_{1}) = (\begin{matrix} - 1 \\ 0 \\ - 1 \\ 0 \\ ⋱ \end{matrix}),$ $Δ_{0}^{2} (T_{2}) = (\begin{matrix} 0 \\ - 1 \\ 0 \\ - 1 \\ ⋱ \end{matrix}),$ are $(m - 1) \times (m - 1)$ matrices with rows and columns indexed by $C_{2}, C_{12}, C_{212}, \dots,$ and $Δ_{0}^{\{1, 2\}} (T_{1}) = (0),$ $Δ_{0}^{\{1, 2\}} (T_{2}) = (0) .$ Hence $Δ_{0} (\emptyset) ≅ L (\emptyset),$ $Δ_{0}^{\{1\}} ≅ \{\begin{array}{rcl} \frac{m - 1}{2} L (\{1\}) \oplus \frac{m - 1}{2} L (\{2\}), & if & m is odd, \\ \frac{m}{2} L (\{1\}) \oplus \frac{m - 2}{2} L (\{2\}), & if & m is even, \end{array}$ $Δ_{0}^{\{1\}} ≅ \{\begin{array}{rcl} \frac{m - 1}{2} L (\{1\}) \oplus \frac{m - 1}{2} L (\{2\}), & if & m is odd, \\ \frac{m - 2}{2} L (\{1\}) \oplus \frac{m - 1}{2} L (\{2\}), & if & m is even, \end{array}$ $Δ_{0}^{\{1, 2\}} ≅ L (\{1, 2\}) .$ So the decomposition matrix for the cell representations is $κ = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \frac{m - 1}{2} & \frac{m - 1}{2} & 0 \\ 0 & \frac{m - 1}{2} & \frac{m - 1}{2} & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) when m is odd,$ and $κ = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \frac{m}{2} & \frac{m - 3}{2} & 0 \\ 0 & \frac{m - 3}{2} & \frac{m}{2} & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) when m is even.$ Thus $κ = Cbb$ for dihedral Hecke algebras at $q = 0$ (but the cell modules t $q = 0$ are not the projective indecomposables - the cell modules at $q = 0$ are semisimple.)

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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