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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119 views

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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94 views

Hecke operators self-adjoint with respect to Petersson inner product

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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94 views

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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28 views

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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1answer
78 views

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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58 views

Conventions for Kazhdan Lusztig-polynomials and the form of the KL-conjecture

This is a follow-up question to Reference request: Presentations of Hecke algebras. In the common definition $$H_t = \langle T_1,\dotsc, T_n \mid \text{braid relations, and for all i: $T_i^2 = t + (t-...
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99 views

Two definitions of the Jacquet functor: why are they the same?

Let $G$ be (the rational points of) a connected, reductive group over a local field $F$. Let $P$ be a parabolic subgroup of $G$ with unipotent radical $N$ and Levi subgroup $M$. The inclusion $M \...
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60 views

$H(G') \otimes_{H(G)} V$ and $f^{-1} \mathcal G \otimes_{f^{-1}\mathcal O_Y} \mathcal O_X$, the connection

1 . Let $f: (X,\mathcal O_X) \rightarrow (Y,\mathcal O_Y)$ be a morphism of ringed spaces, and let $\mathcal G$ be a sheaf of $\mathcal O_Y$-modules. Define the inverse image $f^{\ast} \mathcal G$ ...
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122 views

Directed system of irreducible modules

I have a question on a paper in the Corvallis proceedings on automorphic forms. Background: Let $G$ be a topological group of td type. This means that $G$ is Hausdorff, and every neighborhood of the ...
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1answer
98 views

Isomorphism of Hecke algebra $H(G_1 \times G_2)$ with $H(G_1) \otimes H(G_2)$

Let $G$ be a topological group of totally disconnected (td) type. This means that the identity of $G$ has a fundamental system of neighborhoods consisting of open compact subgroups. Then $G$ is ...
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118 views

Hecke operators acting on the Jacobian $J(X_1(N))$

I'm stuck with p.238 of Diamond's book on modular forms. If $\{f_j\}_{j=1}^g$ is the eigenform basis of $S_2(\Gamma_0(N))$, fixing $p_0 \in X_0(N)$ we have an holomorphic map $$\phi : X_0(N) \to J(X_0(...
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39 views

Finite group with $BN$-pair, if $s_i$ and $s_j$ are conjugate in $W$, why do they have the same index parameter?

Suppose $G$ is a finite group with $BN$-pair, and $(W,S)$ is its Coxeter system. Iwahori's theorem that the corresponding Hecke algebra $\mathcal{H}$ has a standard basis $\{a_w:w\in W\}$ where $a_w=...
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55 views

Why is the index map on the Hecke algebra $\mathcal{H}(G,H,1_H)$ an algebra homomorphism?

Suppose $H\leq G$ are finite groups, $k$ a splitting field for $G$, and $\mathcal{H}(G,H,1_H)=ekGe$ the corresponding Hecke algebra, where $e=\frac{1}{|H|}\sum_{h\in H}h$ is the idempotent ...
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54 views

Why is $e_HKGe_H$ isomorphic to the algebra of functions $G\to K$ constant on $(H,H)$-double cosets?

Suppose $H$ is a subgroup of a group $G$, $K$ a field, $|H|$ is invertible in $K$, and $e_H=|H|^{-1}\sum_{h\in H}h\in KG$. Why is the Hecke algebra $e_HKGe_H$ isomorphic to the algebra of functions $...
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1answer
118 views

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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1answer
333 views

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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52 views

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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32 views

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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42 views

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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42 views

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 ...
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66 views

The development of the Hecke algebras on the representation.

Recently, I am interested in the Hecke algebras, but I'm not very familiar with it. Who can provide me some meterials about the history and the development of the Hecke algebras? Is there any good ...
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55 views

Defining Hecke-Algebra via Generators and Relations

Given a field $K$ and an element $q \in K$ one can define the Hecke-Algebra $H_n(q)$ ($n \geq 2$) by the generators $1, g_1, ..., g_{n-1}$ and the relations (1) ... (2) ... (3) $g_i ^2 = (1-q)g_i +...
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48 views

The decomposition number $D^\mu$ in $C^\lambda$ in the Cellular Algebra $A$.

Consider the Cellular Algebra $A$ with a poset $\Lambda$, for any $\lambda,\mu$ in $\Lambda$, there is a A-module $C^\lambda$ and a irreducible A-module $D^\mu=C^\mu/radC^\mu$. Denote $d_{\lambda\mu}$...
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1answer
27 views

If $M$ is a simple $R$-module, and an $F$-space, why does $End_F(M)\cong M^{\oplus\dim_F(M)}$?

Suppose a ring $R$ is an $F$-algebra for $F$ a field, and $M$ is a simple $R$-module and a finite dimensional $F$-vector space. We can endow $\operatorname{End}_F(M)$ with an $R$-module structure by ...
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1answer
197 views

Degenerations of affine Hecke algebras

Consider an affine Hecke algebra $H$ corresponding to some semisimple algebraic group $G$. Let $H_{deg}$ denote the corresponding degenerate affine Hecke algebra. The algebra $H_{deg}$ can be obtained ...
3
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1answer
116 views

Archimedean Hecke Algebra of $\operatorname{GL}_1(F)$ for $F = \mathbb{R}$ or $\mathbb{C}$

Suppose we have a $\operatorname{GL}_1$ over a number field $F$. I am interested in a description for the archimedean Hecke algebra (always taking the maximal compact subgroup). We know will be the ...
3
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1answer
118 views

Is there a synthetic definition of the $0$-Hecke monoid of $S_n$?

Background. Let $n$ be a nonnegative integer, and let $S_n$ denote the $n$-th symmetric group. The $0$-Hecke monoid $H_0\left(S_n\right)$ is defined to be the monoid given by generators $t_1, t_2, \...