# Questions tagged [hecke-algebras]

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### Reference Request: Jimbo's Proof of Quantum Schur-Weyl Duality

In his seminal 1986 paper "A $q$-analogue of $U(\mathfrak{gl}(N+1))$, Hecke Algebra, and the Yang-Baxter Equation", Jimbo asserted (Proposition 3) that the quantum group associated to $\mathfrak{gl}_n$...
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### Q Exercise 4, Hecke Algebras - Daniel Bump

I'm struggling with exercise 4 in Bump's Stanford Hecke Algebra notes linked here It states the following: Let $G$ be a finite group and $V,W$ vector spaces. Let $C(G,V)$ denote the spaces of maps ...
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I am working through Murty's 'Problems in the Theory of Modular Froms', but I am stuck in the proof of Hecke operators being self adjoint with respect to the Petersson inner product. Before describing ...
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### Iwahori-Hecke algebra of $GL_2$

I had make an old post about this, but since seeing it now shows that I had no idea how to explain my question (because I didn't understand it) I'll try to do it normally here. So I am studying this ...
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### Algebra-module over a set, is there such a thing?

I'm reading a paper and $H$ is an algebra and $M$ is a set. And then they define $M|H$ to be the "$H$-module $M$". Is it obvious what this is or is it bad math description? My guess is the free $H$-...
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### Hecke Bound for Cusp - Modular Forms

The problem statement, all variables and given/known data i have a questions on the piece of lecture notes attached: Relevant equations The attempt at a solution I agree 2) of proposition 2.12 ...
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### Some questions on the Hecke algebra in Casselman's notes

Let $G$ be a connected, reductive group over a $p$-adic field $F$, which is unramified in the sense that $G$ is quasisplit to split over an unramified extension of $F$. Then $G$ is necessarily ...
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### Congruences between Modular forms of different weight

I'm trying to fill in the details of a computation from example 2.4 here on page 12. To set up, consider the spaces of modular forms $M_k(1)$ for weights $k=4,6$. These are both one dimensional and ...
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### Order of sublattice.

I am reading serre's book A Course in Arithmetic"", and want to understand the Hecke operator. Let $\omega_1,\omega_2\in \mathbb{C},$ and $\frac{\omega_1}{\omega_1}\in \mathbb{H},$ where $\mathbb{H}$ ...
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### An equation about Lagrange polynomial in the Yokonuma Hecke algebra

I am reading the paper https://arxiv.org/pdf/1611.03265.pdf Let $\mathcal{J}$ be the commutative subalgebra of the Yokonuma Hecke algebra $Y_{r,n}(q)$ geberated by $t_1,\dots t_n$ which is ...
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### What is the average absolute value if adding a matrix by a random invertible matrix.

Let B be a fixed matrix over non-archimedean field F. Let X be a random invertible matrix over $O_F$(Entries in $O_F$ and The determinant is unit), $O_F$ is the integer ring of F. What is the ...
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### Relations between finite Hecke algebras and affine Hecke algebras.

A finite Hecke algebra $H_n(q)$ of type $A$ is an associated algebra generated by $T_1, \ldots, T_{n-1}$ with relations \begin{align} & T_i T_j = T_j T_i, |i-j|>1, \\ & T_i T_{i+1} T_i = T_{...
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### A question concerning the Cellular Algebra, which is why $(P\otimes_AC^{\mu\star})\otimes_RC^\mu\cong(\dim P\otimes_AC^{\mu\star})C^\mu$?

Let $A$ be a Cellular Algebra over the integral domain $R$. Let $C^{\mu\star}$ be a left $A$-module and $C^\mu$ be a right $A$-module. Let $P=Ae$ be a principal indecomposable $A$-module. What ...