# Questions tagged [hecke-algebras]

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### Has anyone met this "$q$-character" table for $S_4$?

Is anyone aware of the following $q$-character table for the symmetric group $S_4$? \begin{array}{|c|c|c|c|c|c|} \hline \mathrm{conj}\backslash\mathrm{rep} & 2+1+1 & 3+1 & ...
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### Orthogonal similitudes

While considering Hecke theory, a question arised whether there are "irrational" similitudes. The question is easy to formulate: Let S be an even (integral matrix with even diagonals), ...
1 vote
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### Commutative subalgebra of Iwahori-Hecke algebra

I am currently reading the (french) book  by Matsumoto. In (2.1.11) the following statement (reformulated in my own words/notation) appears: Let $(W,S)$ be a Coxeter system, let $R$ be a ...
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### Barr's element and the type-A Iwahori–Hecke algebra

Let $\mathfrak{S}_n$ denote the symmetric group on $n$ letters. Barr's element $\mathcal{S}(n) \in \Bbb{R}\bigl[ \mathfrak{S}_n \bigr]$ is defined as \begin{equation} \mathcal{S}(n) := \ \sum_{i=1}^{n-...
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### Hecke algebras with modified braid relations

I'm studying certain games on graphs on $n$ vertices and it turns out that I get a matrix algebra with generators $T_i , i=1,\dotsc,n$ satisfying a certain type of relations. Here the $T_i$ are ...
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### On the order of the head of product of two simple modules over Quiver Hecke Algebras

My question is: We assume the underlying quiver is a Dynkin quiver. Let $L(\lambda)$ and $L(\mu)$ be two simple modules over Quiver Hecke algebra $R$ where $\lambda$ and $\mu$ are two Konstant ...
1 vote
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### Classical Hecke operators and Hecke algebra of type $A_1$

What's the relation between the classical Hecke operators (as defined in J. P. Serre's A course in arithmetic chapter 6) and the Hecke algebra of type $A_1$, i.e. the algebra generated by the vertices ...
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### Reference request: Finite (multi-parameter) Iwahori-Hecke algebras are pairwise isomorphic

Let $(W,S)$ be a Coxeter system. Let $q=(q_s)_{s\in S} \in \mathbb{R}^{\text{#}S}$ be a tuple of positive real numbers with $q_s=q_t$ whenever $s$ and $t$ are conjugate to each other. Follwing Davis', ...
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### On the status of some conjectures mentioned/used in Harish Chandra's 1970 lecture notes

In van Dijk's notes of Harish Chandra's lectures on harmonic analysis, several conjectures are mentioned throughout, such as in Part 1, section 4 of van Dijk's notes Conjecture I : Let $\omega$ be ...
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### Homogenous Hermitian form on the KLR algebra

Does there exist a homogenous conjugate-linear automorphism of the KLR algebra? I want to be able to define a homogenous Hermitian form on the (Specht) modules of the (cyclotomic) KLR algebra.
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### Flag variety over quaternions and its Hecke algebra

Consider cell decomposition of flag variety of $\mathbb{H}^{n}$ into orbits under the action of $B(\mathbb{H})$ - group of upper triangular matrices with coefficients in $\mathbb{H}$. I think cells ...
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### Volumes of double cosets $KtK$

Let $G$ be the group of rational points of a connected reductive group $\mathbb G$ defined over a non-archimedean local field $F$. For simplicity sake, I assume that $\mathbb G$ is split over $F$. Let ...
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### Smallest Mazur's good prime

Let $p$ and $\ell$ be primes $\geq 5$ such that $\ell$ divides $p-1$. Following Mazur, we say that a prime $q$ is a $\textit{good prime}$ if $\ell$ does not divide $q-1$ and $q$ is not a $\ell$th ...
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### Admissible (unitary) spherical representation $sl(2,Q_p)$. Does dimension fixed point vector increase proportional index

I am using terminology of Cartier's Harmonic analysis on trees. Take $\pi$ be one of the irreducible principal or complementary (unitary) spehrical series of $Sl(2, Q_p)$. Let $K$ be the maximal ...
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### Basic theorem on induction for representations of $p$-adic groups

I know a lot of places where the following is sparsely proved, but I remember there was some paper where I read it in basically the same form I write it, but unfortunately I can't remember where it ...
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### Why are the definitions of i-good nodes of a multipartition equivalent?

Let $e\geq 2$ and $0\leq i\leq e-1$. For a multipartition $\lambda\vdash_\ell n$ of $n$ one can define the notion of an $i$-good box of the Young diagram of $\lambda$. But there are seemingly ...
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### Hecke algebra of GL(2,F)

I was studying about the Hecke algebra from Bernstein's notes on p-adic representation theory and various other sources. First a disclaimer: everything below is fairly new to me so please feel free to ...
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### What is the kernel of the action of the Iwahori-Hecke algebra?

The Iwahori-Hecke algebra $H_n(q)$ acts on the $n$th tensor power of the standard representation of $U_q(\mathfrak{sl}_m)$. What is the kernel of this action? Does anyone know a reference? I'm happy ...