The Iwahori-Hecke algebra of the group <math> <mi>G</mi><mfenced> <mi>r</mi><mi>r</mi><mn>2</mn> </mfenced> </math> of order <math> <mn>2</mn><mi>r</mi> </math>

The Iwahori-Hecke algebra of the group $G (r, r, 2)$ of order $2 r$

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: 28 May 2010

The Iwahori-Hecke algebra of the group $G (r, r, 2)$ of order $2 r$

Let $p, q \in ℂ^{*} .$ The Iwahori-Hecke algebra of the dihedral group $H_{r, r, 2} = H_{r, r, 2} (p, q)$ is the algebra given by the generators $T_{1}, T_{2}$ and relations $(T_{1} T_{2} T_{1} T_{2} \dots (r factors)) = (T_{2} T_{1} T_{2} T_{1} \dots (r factors)) T_{1}^{2} = (p - p^{- 1}) T_{1} + 1, T_{2}^{2} = (q - q^{- 1}) T_{2} + 1 .$ The last two relations guarantee that $T_{1}$ and $T_{2}$ are invertible. $T_{1}^{- 1} = T_{1} - (p - p^{- 1}) and {T_{2}}^{- 1} = T_{2} - (q - q^{- 1}) .$ If $r$ is odd then $p = q$ is forced since ${(T_{1} T_{2} T_{1} T_{2} \dots (r - 1 factors))}^{- 1} T_{2} (T_{1} T_{2} T_{1} T_{2} \dots (r - 1 factors)) = T_{1} .$ The elements $\{1, (T_{1} T_{2} T_{1} T_{2} \dots (k factors)), (T_{2} T_{1} T_{2} T_{1} \dots (k factors)), (T_{1} T_{2} T_{1} T_{2} \dots (r factors)) | 1 \leq k < r\}$ form a basis of $H_{r, r, 2} .$ If $p = q = 1$ then $H_{r, r, 2} = ℂ G (r, r, 2)$ is the group algebra of the dihedral group of order $2 r .$

Assume that $p, q$ satisfy ?????. The irreducible representations of $H_{r, r, 2} (p, q)$ are given as follows. The one dimensional representations $ρ^{+ +}$ and $ρ^{- -}$ are given by $ρ^{+ +} (T_{1}) = p, and ρ^{+ +} (T_{2}) = q,$ and $ρ^{- -} (T_{1}) = - p^{- 1} and ρ^{- -} (T_{2}) = - q^{- 1},$ and, if $r$ is even, there are additional one dimensional representations $ρ^{+ -}$ and $ρ^{- +}$ given by $ρ^{+ -} (T_{1}) = p, and ρ^{+ -} (T_{2}) = - q^{- 1},$ and $ρ^{- +} (T_{1}) = - p^{- 1} and ρ^{- +} (T_{2}) = q .$ The two dimensional representations $ρ^{λ}, 0 < λ < r / 2$ are given by $ρ^{λ} (T_{2}) = (\begin{matrix} a & (1 + a d) \\ 1 & d \end{matrix}) and ρ^{λ} (T_{2}) = (\begin{matrix} - d ξ & (1 + a d) ξ \\ ξ^{- 1} & - a ξ^{- 1} \end{matrix}) where ξ = e^{2 π i / r},$ $a = \frac{(q - q^{- 1}) ξ + (p - p^{- 1})}{ξ - ξ^{- 1}}, and d = \frac{(q - q^{- 1}) ξ^{- 1} + (p - p^{- 1})}{ξ^{- 1} - ξ} .$

Proof.

Instead of repeating the proof of the theorem in the previous section and minding our p's and q's (SLAP! -NS) let us give a proof showing how the result can be derived.

The quadratic relations in ????? are equivalent to $(T_{1} - p) (T_{1} + p^{- 1}) = 0 and (T_{2} - q) (T_{2} + q^{- 1}) = 0,$ and these imply that any one dimensional representation $χ : H_{r, r, 2} \to ℂ$ must have $χ (T_{1})$ equal to $p$ or $p^{- 1}$ and $χ (T_{2})$ equal to $q$ or $q^{- 1} .$ If $r$ is odd then $T_{1}$ is conjugate to $T_{2}$ and so $χ (T_{2}) = χ (T_{1}) .$ This determines our one dimensional representations of $H_{r, r, 2} .$

Let $ρ : H_{r, r, 2} \to M_{2} (ℂ)$ be an irreducible two dimensional representation of $H_{r, r, 2} .$ By the relation $(T_{1} - p) (T_{1} + p^{- 1}) = 0$ be the eigenvalues of $ρ (T_{1})$ are in the set $\{p, p^{- 1}\} .$ If $ρ (T_{1})$ has two equal eigenvalues then it is a multiple of the identity. If $v$ is an eigenvector of $ρ (T_{2}),$ and $ρ (T_{1})$ is a constant multiple of the identity then $ℂ v$ is a submodule of $ρ$ and this contradicts the irreducibility of $ρ .$ So $ρ (T_{1})$ must have eigenvalues $p$ and $p^{- 1} .$ Similarly $ρ (T_{2})$ must have eigenvalues $q$ and $q^{- 1} .$

By the defining relations, $T_{1} (T_{1} T_{2} \dots (m factors)) = T_{1} (T_{2} T_{1} \dots (m factors)) = (T_{1} T_{2} \dots (m factors)) T_{i},$ where $i = 1$ if $r$ is odd and $i = 2$ if $r$ is even. So $T_{2} {(T_{1} T_{2})}^{m} = T_{1} (T_{1} T_{2} \dots (m factors)) (T_{1} T_{2} \dots (m factors)) = (T_{1} T_{2} \dots (m factors)) T_{i} (T_{1} T_{2} \dots (m factors)) = (T_{1} T_{2} \dots (m factors)) (T_{1} T_{2} \dots (m factors)) T_{1} = {(T_{2} T_{1})}^{m} T_{1},$ and simiilarly one shows that $T_{2} {(T_{1} T_{2})}^{m} = {(T_{1} T_{2})}^{m} T_{2} .$ So ${(T_{1} T_{2})}^{m}$ is in the center of $H_{r, r, 2} .$

By Schur's lemma, $ρ {(T_{1} T_{2})}^{m} = c Id, for some c \in ℂ .$ Since $ρ (T_{1})$ has eigenvalues $p$ and $p^{- 1}$ and $ρ (T_{2})$ has eigenvalues $q$ and $q^{- 1},$ $\det (ρ (T_{1})) = \det (ρ (T_{2})) = - 1$ and so $\det (ρ {(T_{1} T_{2})}^{m}) = {(- 1)}^{2 m} = 1 .$ On the other hand, $\det (ρ {(T_{1} T_{2})}^{m}) = c^{2}$ and so $c = \pm 1 .$ So $ρ {(T_{1} T_{2})}^{m} = c Id .$ Thus the eigenvalues of $ρ (T_{1} T_{2})$ are $r$ th roots of unity. Since $\det (ρ (T_{1} T_{2})) = {(- 1)}^{2} = 1,$ we can write (possibly after some change of basis in the module $ρ$ ) $ρ (T_{1} T_{2}) = (\begin{matrix} ξ^{k} & 0 \\ 0 & ξ^{- k} \end{matrix}) where ξ = e^{2 π i / r} and k \in ℤ .$ If $k = 0$ or (if $r$ is even and) $k = r / 2$ then $ρ (T_{1} T_{2}) = \pm 1$ and $ρ$ is not invertible. So without loss of generality we can assume that $0 < k < r / 2 .$

Suppose that $ρ (T_{2}) = (\begin{matrix} a & b \\ c & d \end{matrix}) .$ The relation $T_{2}^{2} = (q - q^{- 1}) T_{2} + 1$ implies that $a + d = q - q^{- 1}$ and $a d - b c = 1 .$ So we can assume $ρ (T_{2}) = (\begin{matrix} a & a d + 1 \\ 1 & d \end{matrix}), with a + d = q - q^{- 1} .$ Then $ρ (T_{1}) = ρ (T_{1} T_{2} {T_{2}}^{- 1}) = ρ (T_{1} T_{2}) ρ {(T_{2})}^{- 1} = (\begin{matrix} ξ^{k} & 0 \\ 0 & ξ^{- k} \end{matrix}) (\begin{matrix} - d & a d + 1 \\ 1 & - a \end{matrix}) = (\begin{matrix} - d ξ^{k} & (1 + a d) ξ^{k} \\ ξ^{- k} & - a ξ^{- k} \end{matrix}) .$ Now $Tr (ρ (T_{1})) = p - p^{- 1} = - d ξ^{k} - a ξ^{- k}, and Tr (ρ (T_{2})) = q - q^{- 1} = a + d,$ and these equations determine $a$ and $d$ as given in the statement of the theorem. $□$

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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The Iwahori-Hecke algebra of the group G rr2 of order 2r

The Iwahori-Hecke algebra of the group G rr2 of order 2r

References [PLACEHOLDER]

The Iwahori-Hecke algebra of the group $G (r, r, 2)$ of order $2 r$

The Iwahori-Hecke algebra of the group $G (r, r, 2)$ of order $2 r$