All Questions
Tagged with hecke-algebras rt.representation-theory
74 questions
2
votes
0
answers
92
views
Geometric interpretation of flags and the role of the rook monoid and Kazhdan–Lusztig theory in $M_n(\mathbb{C})$
Let $G = GL_n(\mathbb{C})$, $B$ be its Borel subgroup, and $P$ a parabolic subgroup. The space $G/B$ corresponds to complete flags in $ \mathbb{C}^n$, and $G/P$ corresponds to partial flags. The ...
- 141
11
votes
1
answer
331
views
A question on groups having a subgroup which fixes a vector in every irreducible representations
Given a finite group $G$, I am interested in finding a non-trivial proper subgroup $H$ of $G$ such that $\mathrm{Ind}_H^G\mathbf{1}$ contains all the irreducible representations of $G$, that is, ...
- 251
2
votes
0
answers
61
views
Quiver and relations for the 0-Hecke algebra
Let $A_n=H_n(0)$ denote the 0-Hecke algebra, see for example König - $0$-Hecke algebras of the symmetric groups.
All simple $A_n$ modules are 1-dimensional and thus $A_n$ is isomorphic to an algebra ...
- 26.5k
3
votes
0
answers
72
views
How to multiply dots with Young idempotents in the degenerate affine Hecke algebra (type A)
Let $\widehat{\cal H}_n$ be the type A degenerate affine Hecke algebra on $n$ strands, and let $x_1,\cdots,x_n$ be the dots. Inside of this algebra lies the algebra $\mathbb C S_n$, and the Young ...
- 311
4
votes
0
answers
110
views
Hecke algebra $\mathcal{H}(K_1\backslash \mathrm{GL}_n(\mathbb{F})/K_1)$
$\DeclareMathOperator\GL{GL}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers, and let $\frak{m}$ be its maximal ideal Let $\GL_n(\mathcal{O})$ be the group ...
- 21.8k
1
vote
0
answers
100
views
Automorphisms of Iwahori/affine Hecke algebras
Has there been any serious study of automorphisms of extended affine Hecke algebras? Has anyone determined the automorphism group of say, type A extended affine Hecke algebras? I ask because the ...
5
votes
2
answers
429
views
Parabolic induction, and tensoring (Iwahori/affine) Hecke algebras
In several places, such as [1] and [2], it seems to be implicitly known that (normalized) parabolic induction corresponds to tensoring affine Hecke algebras. I say implicitly because both [1] and [2] ...
4
votes
1
answer
230
views
Eigenvalue of Iwahori Hecke Algebra element for the Steinberg
In Iwahori-Matsumoto's paper the Iwahori Hecke Algebra for $G=GL_n(F)$ is generated by $X_{s_0}, X_{s_i},i\in\{0,...,n-1\}$ and $ X_{\rho}$ with the relations:
$
1) (X_{s_{i}}-q)(X_{s_{i}}+1)=0\:,\;\;...
- 141
4
votes
1
answer
161
views
Spectral projection of an eigenvalue associated to a generator of Hecke algebras
In his paper "Hecke Algebras of type $A_n$ (Inv. Math. 1988, EUDML link) and subfactors", in section 2 "Orthogonal representations...", Wenzl takes the usual third relation of a ...
- 101
1
vote
0
answers
64
views
Admissibility of representations induced from Hecke algebra for covering groups
Assume $G$ is a semisimple algebraic group and $B$ is an Iwahori subgroup. Let $(r,E)$ be a representation of $H(G,B)$ which is an Iwahori-Hecke algebra, then Borel proved that $C_{c}(G/B)\otimes_{H}E$...
- 89
1
vote
1
answer
190
views
Commutative subalgebra of Iwahori-Hecke algebra
I am currently reading the (french) book [1] 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 ...
- 741
7
votes
0
answers
272
views
Naive Schur-Weyl duality for the 0-Hecke algebra
The 0-Hecke algebra $\mathcal{H}_n(0)$ is the $\Bbb{C}$-algbra generated by elements $T_1,
\dots, T_{n-1}$ satisfying the braid relations
and the idempotency relations $T_i^2 = T_i$.
It is known that $...
- 2,137
3
votes
0
answers
136
views
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-...
- 2,137
4
votes
0
answers
76
views
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 ...
- 71
1
vote
0
answers
57
views
Block sum for degenerate affine Hecke algebras
The degenerate affine Hecke algebra $H_k$ over a field $F$ is the algebra with generators $s_1,\ldots,s_{k-1}$ and $x_1,\ldots,x_k$, subject to the following relations:
$s_is_js_i=s_js_is_j$ for $i=j\...
- 721
3
votes
0
answers
103
views
compactly induction of smooth modules over Hecke algebras
Let $G$ be a locally profinite group and $H$ its closed subgroup. It is well-known that a smooth representation of $G$ is identified with a smooth module over $\mathcal{H}(G)$, where $\mathcal{H}(G)$ ...
- 479
13
votes
0
answers
207
views
Hidden grading on $kS_n$
Brundan and Kleshchev showed that if $k$ is a field of characteristic $p$, then the group ring $kS_n$ of the symmetric group $S_n$ admits an integer grading which is nontrivial in the sense that it is ...
- 721
10
votes
3
answers
419
views
Positivity of Iwahori–Hecke algebra characters on the Kazhdan-Lusztig basis
$\DeclareMathOperator\tr{tr}$I'm interested in the irreducible characters of a finite Iwahori–Hecke algebra evaluated at the Kazhdan–Lusztig basis. These are Laurent polynomials.
Are the coefficients ...
- 617
6
votes
1
answer
210
views
Positivity of Schur elements in Iwahori-Hecke algebras
I'm interested in finite Iwahori-Hecke algebras.
If $\mathcal{H}$ is such a Hecke algebra, defined over $\mathbb{Z}[q^{\pm 1/2}]$, and $\Lambda$ an irreductible representation, there is the notion of ...
- 617
8
votes
3
answers
635
views
Motivation for the Kazhdan-Lusztig involution
I would like to know about the motivation behind the Kazhdan–Lusztig involution on an Iwahori–Hecke algebra.
I'll borrow the conventions from Libedinsky's Gentle introduction to Soergel bimodules I: ...
- 721
7
votes
2
answers
944
views
Reference of J.L. Waldspurger's paper on Shimura correspondence
I want to find reference of Waldspurger's paper referred at "Sur les coefficients de Fourier des formes modulaires de poids demi-entier" J. Math. Pures Appl. (9) 60 (1981), no. 4, 375–484 (...
- 73
1
vote
0
answers
52
views
Promoting representation of a subgroup to representation of an action groupoid
Suppose I have a group $G$, a subgroup $K \leq G$, and a representation $(\sigma, V)$ of $K$.
There is a natural left action of $G$ on $X := G/K \times G/K$ given by $g \cdot (g'K,g''K) = (gg'K,gg''K)$...
- 503
8
votes
1
answer
406
views
Structure constants for the double coset algebra of a Young subgroup
Fix a Young subgroup $H_\lambda \subseteq \mathcal S_n$, where $\lambda \vdash n$ is a partition of $n$ with $k$ blocks. Inside the group algebra $\mathbb C[\mathcal S_n]$, consider the idempotent
$$\...
- 418
8
votes
1
answer
249
views
Hecke algebra relation versus $\operatorname{SL}_2$ trace relation
The quadratic relation in the (type $A$) Hecke algebra is $(T-t)(T+t^{-1}) =0$, which can be rewritten as
$$
T-T^{-1} = t-t^{-1}$$
Suppose $A \in \operatorname{SL}_2(\mathbb{Q})$ with eigenvalues $a,...
- 2,869
3
votes
0
answers
147
views
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 ...
- 5,230
3
votes
1
answer
237
views
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', ...
- 741
9
votes
0
answers
237
views
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 ...
- 113
4
votes
1
answer
355
views
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 ...
- 6,349
2
votes
0
answers
67
views
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 ...
- 193
5
votes
2
answers
545
views
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 ...
- 463
1
vote
0
answers
40
views
Decomposition numbers of characteristic 0 Ariki-Koike algebras for q=0 or q generic
Consider a complex Ariki-Koike algebra $H:=H(q;Q_1,\dots, Q_\ell)$ for some invertible elements $q, Q_1,
\dots, Q_\ell$ of $\mathbb{C}$, i.e. an associative $\mathbb{C}$-algebra with generators $T_0, \...
2
votes
0
answers
55
views
Branching rule for degenerate cyclotomic Hecke algebras
Grojnowski and Vazirani showed that there is a crystal isomorphism between the crystal defined by modular branching of simple modules of Ariki-Koike algebras and that of an integral highest weight ...
4
votes
1
answer
350
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$-...
1
vote
0
answers
42
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 ...
18
votes
2
answers
1k
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 ...
- 6,170
4
votes
1
answer
241
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}{...
9
votes
2
answers
1k
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 ...
7
votes
0
answers
202
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 ...
- 425
5
votes
1
answer
716
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 ...
- 5,424
4
votes
0
answers
241
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 ...
- 2,809
3
votes
0
answers
234
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 ...
- 6,201
1
vote
0
answers
32
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 ...
- 331
7
votes
1
answer
185
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 ...
- 6,201
3
votes
0
answers
414
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 ...
- 331
1
vote
0
answers
128
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 ...
- 6,201
2
votes
0
answers
150
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 ...
- 2,273
4
votes
1
answer
355
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 ...
- 2,273
4
votes
0
answers
202
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\...
- 868
7
votes
1
answer
357
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 ...
- 3,830
8
votes
1
answer
681
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 ...
- 85.6k