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2
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49 views

Basis for Annular Skein Algebra

Background/Notation: Given the Iwahori-Hecke algebra $H_{n}$ (over some ring commutative ring $R$ with identity) with generators $\{T_{1},\ldots T_{n-1}\}$, we know it has basis $\{T_{w}\}_{w\in S_{...
4
votes
1answer
181 views

Examples of non-trivial Kazhdan-Lusztig polynomials

I'm looking for examples of non-trivial Kazhdan-Lusztig polynomials, specifically in the case where the Coxeter system is a Weyl group. For example, the simplest polynomial with non-trivial $q$-...
3
votes
1answer
86 views

Categorifying skein algebras?

We can obtain the Jones polynomial by the Temperly-Lieb algebra and the HOMFLYPT polynomial from the Hecke algebra. Were there attempts to categorify the algebras itself and obtain the Khovanov ...
3
votes
1answer
149 views

Understanding the Hecke Algebra via Different Constructions

I'm reading through this paper by Pouchin, and in it he makes a clam that some function space is isomorphic to the Hecke algebra. I'm trying to understand this and could really use some help. Let $G =...
1
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0answers
31 views

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 ...
14
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2answers
729 views

What is the archimedean Hecke algebra?

Let $\mathbf G$ be a connected, reductive group over $\mathbb Q$. For each nonarchimedean place $v$, let $K_v$ be a maximal compact subgroup of $\mathbf G(\mathbb Q_v)$. The space $\mathscr H(\mathbf ...
2
votes
0answers
91 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 ...
4
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1answer
166 views

Number of pairs of permutation in $S_n$ whose $\mu$-coefficient (of their Kazhdan Lusztig polynomial) is non-zero

I am interested in how many pairs of permutations $(u,w)$ in $S_n$, such that the $\mu$-coefficient of its Kazhdan-Lusztig polynomial $P_{u,w}(q)$ is non-zero? where $\mu_{u,w}=[q^{\frac{l(w)-l(u)-1}{...
6
votes
2answers
308 views

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 ...
6
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0answers
89 views

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 ...
4
votes
1answer
318 views

Relation between Hecke operators and coefficient of L-functions

This question has its seed in this one by Gory, which found an enlightening answer but one of the comments kept me wondering. I am beginning to discover Hecke operators, and there appears to be an ...
20
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0answers
282 views

A spin extension of a Coxeter group?

Consider a Coxeter group $W$ with simple generators $S$ and Coxeter matrix $\left( m_{s,t}\right) _{\left( s,t\right) \in S\times S}$. Let $\mathfrak{M}$ be the set of all pairs $\left(s, t\right) ...
4
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0answers
123 views

Semisimple vs ordinary trace of Frobenius on nearby cycles of affine flag variety

Let $G \rightarrow X$ be a parahoric group scheme over a curve, with parahoric level structure at $x_0$. Gaitsgory essentially showed that the nearby cycles functor $R\Psi$ takes perverse sheaves on ...
2
votes
1answer
118 views

Krull dimension of Hecke algebra (level 1) for p = 2, 3

Let $p$ be a prime and consider the ($p$-deprived) Hecke algebra $\mathbb{T}$ which the projective limit of the Hecke $\mathbb{Z}_p$-algebras $\mathbb{T}_k$ which act on modular forms of level $1$ ...
2
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0answers
45 views

Reference request: Hecke agebra over non-commutative rings

I think the title sums it up quite well: Is it a useful idea to define the Iwahori-Hecke algebra over a non-commutative $k$-algebra? If so, what shape should the relations attain? Bonus question: ...
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0answers
40 views

symmetric polynomials for Super Hecke Clifford algebra

Fix a natural number $n$. In https://arxiv.org/abs/1107.1039, §3.5, Kang/Kashiwara/Tsuchioka define a (version of a) Hecke Clifford superalgebra. It is the superalgebra with the following generators: ...
2
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1answer
194 views

Noncommutative cohomology of flag varieties

Consider the Grassmannian $Gr(n, N)$ of $n$-dimensional subspaces of $\mathbf C^N$. Its cohomology ring is isomorphic to $\mathbf C[x_1, \ldots, x_n, \bar x_1, \ldots \bar x_{N-n}]/I_{n,N}$, wherethe ...
1
vote
1answer
72 views

About left cell of a permutation

I am reading a paper Cellular algebras by J.J. Graham, G.I. Lehrer. I do not understand the follwing words labelled by yellow. First, I know Robinson-Schensted correspondence of a permutation in the ...
12
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0answers
250 views

Modularity of endomorphism algebras

This question is about comparing Hecke algebras and endomorphism algebras. Let $\mathbf{A}_f$ be the ring of finite adèles of $\mathbf{Q}$ and let $K$ be a compact open subgroup of $\mathrm{GL}_2(\...
2
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0answers
136 views

Quantum Schur-Weyl duality for quantum affine algebras of other types

In the paper by Chari and Pressley, it is proved that the there is functor from the category $C_m$ of finite dimensional representations of the affine Hecke algebra of $GL(m)$ to the category $D_n$ of ...
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0answers
28 views

Cellular basis of $KW(B_2)$

Take $K$ a field. Let $W(B_2)$ be the coxeter group of type $B_2$ with a set $\{s_0,s_1\}$ of generators. Then $W(B_2)=\{1, s_0,s_1,s_0s_1,s_1s_0,s_0s_1s_0, s_1s_0s_1,s_1s_0s_1s_0\}$. I know that the ...
3
votes
1answer
229 views

Does Gorensteinness of $\mathbb{T}_{\mathfrak{m}}$ imply multiplicity one?

Let $N$ be a prime and let $\mathbb{T} \subset \mathrm{End}(J_0(N))$ be the Hecke algebra generated over $\mathbb{Z}$ by $U_N$ and the operators $T_p$ for primes $p \nmid N$. Fix a maximal ideal $\...
5
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0answers
102 views

How to translate multi-segments to Drinfeld polynomials?

Let $\hat{H}_m=\hat{H}_m(q)$ be the Iwahori-Hecke algebra of $GL_m$, see for example, Section 2. The simple $\hat{H}_m$-modules are parametrized by Zelevinsky's multi-segments, See Section 2.2 of the ...
3
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0answers
206 views

Kazhdan-Lusztig basis of the symmetric group algebra $\mathbb{K}S_n$

Let $S_n$ be the symmetric group on the set $\{1,2,\ldots,n\}$ and $\mathbb{K}$ a field. Denote by $\mathbb{K}S_n$ the symmetric group algebra. What's Kazhdan-Lusztig basis of $\mathbb{K}S_n$? I ...
0
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0answers
168 views

Eigenvariety and Hecke algebra

Let $h^{n,ord}(Np^\infty)$ be the cuspidal nearly ordinary Hecke algebra of tame level $N$. For $N \geq 4$, we know that the Hecke algebra is the generic fibre of the Hecke-Hilbert Eigenvariety and so ...
7
votes
1answer
486 views

Deligne-Scholl's motives for modular forms: Hecke operators vs. transposed Hecke operators

EDIT: I moved my original question down to the bottom. The question here at the top is related, at least I suppose that the same phenomenon is behind both of them. In the article "Valeurs de ...
2
votes
1answer
205 views

Restriction to the diagonal of Hilbert eigenforms

Do you know of any reference that discusses whether the restriction to the diagonal of a Hilbert eigenform is an (elliptic) eigenform?
4
votes
2answers
277 views

Is $(G\rtimes H,H)$ a Gelfand pair iff $G$ is abelian?

Let $G$ be a finite group and $K \subset G$ a subgroup. Then $(G,K)$ is a Gelfand pair if the double coset Hecke algebra $\mathbb{C}(K \backslash G / K)$ is commutative. Let $H$ be a subgroup of $...
1
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0answers
106 views

References of an operator $T: V \otimes V \to V \otimes V$

Let $V$ be a vector space with a basis $v_1, \ldots, v_n$ and let $X_{ij} = v_i \otimes v_j$. Then $X_{ij}, i,j=1,\ldots, n$, is a basis of $V \otimes V$. Let $T: V \otimes V \to V \otimes V$ be the ...
2
votes
0answers
131 views

Finite-dimensional representations of DAHA of rank 1

DAHA of rank 1 is defined by the relation $$ (T - t^{1/2})(T + t^{-1/2})=0~, \quad TXT=X^{-1}~, \quad TY^{-1}T=Y~, \quad Y^{-1}X^{-1}YXT^2q^{1/2}=1 .$$ To understand its representations, it is useful ...
3
votes
1answer
262 views

p-adic modular forms, Hecke algebra, deformation theory and modular curves.

Let $h^{ord}(N,\mathcal{O})$ be the $p$-ordinary Hecke algebra, and $\mathfrak{m}$ be a maximal ideal of the semi local ring $h^{ord}(N,\mathcal{O})$ corresponding to a residual representation $\bar{\...
4
votes
1answer
234 views

Finite-dimensional representations of DAHA

It is shown by Berest-Etingof-Ginzburg that there exist finite-dimensional irreducible representations of rational Cherednik algebra $H_c(S_n)$ of $A_{n-1}$ type if and only if the deformation ...
3
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0answers
109 views

Shifts in the decomposition of Bott-Samelson bimodules

Let $k$ be an algebraically closed field of characteristic $0$, let $V=k^n$ be a $k$ vector space of dimension $n$, and let $R=k[V]$ be the ring of polynomial functions on $V$. Suppose that $W\subset\...
5
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0answers
173 views

Is there a geometric interpretation of the Hecke algebra of a Weyl group?

The Weyl group $W$ can be realized as a fundamental group of a topological space. So that its group algebra $\mathbb{Z}(W)$ has a geometric meaning. On the other hand, the Hecke algebra is a ...
4
votes
1answer
290 views

Reference request for $pro-p$ Iwahori subgroup of $GL_n(F)$

I am searching for a book/lecture notes/articles where I can find the definition and properties of the $pro-p$ Iwahori subgroup of $GL_n(F)$,(with examples if possible) the Iwahori decomposition of ...
5
votes
1answer
248 views

Smoothness of Hecke algebras

First I will introduce some notation and definitions. Fix a level $N$ (take $N=1$ if it makes things easier) and a prime $p$. Let $k$ be a finite field of characteristic $p$ and let $\mathcal{C}$ be ...
9
votes
1answer
448 views

Arithmetic Points are Dense on a Hida Family

I am reading the paper "Constancy of the Adjoint L-invariant" by H. Hida (http://www.math.ucla.edu/~hida/ConstP.pdf). Correct me if I'm wrong, but I've read/heard that the arithmetic points $p \in ...
6
votes
1answer
280 views

Do Iwahori-Hecke algebras come from cohomology classes?

Let $W$ be a Coxeter group. The Iwahori-Hecke algebra $H_q(W)$ is a deformation of $k W$. Question: is there some way to interpret the deformation $H_q(W)$ as a cohomology class? It doesn't seem ...
5
votes
1answer
163 views

The (Hecke) double coset von Neumann algebra

It it well-known in the von Neumann algebra theory that for $\Gamma$ a non-trivial countable group, the von Neumann algebra $L(\Gamma)$ generating by $\Gamma$ acting by left multiplication on $l^2(\...
15
votes
1answer
572 views

How many traces are there on Temperley-Lieb, Fuss-Catalan, Iwahori-Hecke, Birman-Wenzl-Murakami-Kauffman, … algebras?

There is a theorem (I believe by Ocneanu) that the Markov trace on the tower of Temperley-Lieb algebras is (essentially) unique. What about just traces on separate algebras? That is, take one of them,...
5
votes
0answers
100 views

divisibility by Bernoulli numbers of discriminant of Hecke algebra over the space of modular forms of level 1

For the space of modular forms and the space of cusp forms (here I only care about the level $1$ case), we have the action by Hecke algebras. Therefore, we can calculate the discriminant of this ...
7
votes
1answer
466 views

Mod 2 modular forms in levels 5 and 25--how to account for this Hecke isomorphism?

The space $P1$ of my earlier question 203755 "Two spaces attached to mod 2 level 9 modular forms...", is essentially the space of mod 2 level 3 modular forms. That such a space should appear inside ...
3
votes
1answer
301 views

Two spaces attached to mod 2 level 9 modular forms--a conjectural Hecke isomorphism

MOTIVATION Nicolas and Serre have analyzed the structure of the space of mod $2$ modular forms of level $1$, viewed as a "Hecke-module". They show that for each $p>2$, the operator $T_p$ acting on ...
4
votes
0answers
83 views

Highest (short) roots and commutation relations in (twisted) DAHA

I am trying to understand, explicitly, the commutation relation between $X_\vartheta$ and $Y_\vartheta = T_0T_{s_\vartheta}$ in the (twisted) DAHA for a root system $R$, where $\vartheta$ is the ...
7
votes
1answer
513 views

Proving that the Jones polynomial is q-holonomic

The Jones polynomial is known to have many different interpretations or definitions, by now. There are connections with QFT, quantum groups, Hilbert schemes, Cherednik algebras, etc. My question is ...
4
votes
1answer
521 views

What does the defect of a block measure?

In the context of decomposition matrices for Hecke algebras of finite Coxeter groups at a root of unity (such as the tables at the end of the book "Hecke algebras at a root of unity" by Geck-Jacon or ...
2
votes
1answer
326 views

Newform and Galois representation (Shimura-Deligne Reciprocity Law)

Shimura-Deligne Reciprocity Law implies that to a newform $f \colon =f(z) \in S^{\mathrm{new}}_k(\Gamma_0(N))$ of weight $k$, one can associate Galois representation $\rho_{f,\lambda} \colon {\mathrm{...
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0answers
151 views

Order of individual Fourier coefficient of a Maass form

Let $D$ be a definite quaternion division algebra over $\mathbb{Q}$ and $\mathcal{O}$ be an Eichler order of $D$. Let $F$ be a Maass form in $L^2(PGL_2(\mathcal{O})\backslash PGL_2(D\otimes\mathbb{R}))...
11
votes
0answers
266 views

The Markov trace via Bott-Samelson fibers?

Let $H_n$ be the Hecke algebra of GL(n), i.e., the algebra over $\mathbb{Q}(q)$ with generators $T_1, \ldots, T_{n-1}$ which satisfy the braid relations and also $T^2 = (q-1) T + q$. Recall the ...
4
votes
2answers
316 views

Gelfand pair and double coset decomposition

Let $F$ be a non-Archimedean local field with ring of integers $O$, $\pi$ be a uniformizer. Let $\tilde{G}$ be a connected algebraic group over $F$ and splits over $F$, fix a split maximal torus $\...