Questions tagged [rt.representation-theory]

Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.

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Definition 2.3 in Lusztig's "Character Sheaves I"

About Definition 2.3 in Character Sheaves I: We have the usual $G$, $B$, $T$, Weyl group $W$ and roots $R$. Lusztig wrote down a definition of $R_{\mathcal{L}}$ as: $$R_{\mathcal{L}}=\{\alpha\in R\;|\;...
Cheng-Chiang Tsai's user avatar
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2 answers
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$G\cong C_4\times A_5$ or $C_2\times C_2\times A_5$?

Let $G$ be a finite group of order $240$. If $G\cong C_4\times A_5$ or $C_2\times C_2\times A_5$, then the all degrees of irreducible $\mathbb{C}$-characters of $G$ are $ [1,1,1,1,~3,3,3,3,3,3,3,3, ~...
C. Simon's user avatar
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$Ext_A^1(J,J)$ for the Jacobson radical $J$ of an algebra $A$

Let algebras be finite dimensional over a field $K$ and let $J$ denote the Jacobson radical (this is the intersection of all maximal right ideals) of an algebra. Being hereditary means that the ...
Mare's user avatar
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Relation between Lie group characters and spherical functions on symmetric spaces

Setup: Let $G/K$ be an irreducible compact Riemannian symmetric space, where $G$ is a simply connected compact real Lie group, and $K$ is the maximal compact connected subgroup of some noncompact real ...
Gro-Tsen's user avatar
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Group schemes and Hyperspecial maximal compact subgroups

Let $F$ be a number field. For each non-archimedean place $v$ let $O_v$ denote the ring of integers. Let $G$ be a connected linear algebraic group defined over $F$. Consider the set of sequences $(K_v)...
Mehta's user avatar
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On constructible Hall algebra and instantons

I heard in a talk by Yan Soibelman that by starting with a quiver $Q$ with a set of vertices $I$ we can either symmetrize or anti-symmetrize its Euler-Ringel form $\chi_Q$. He claims that anti-...
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Restriction of discrete series

QUESTION Let $G$ be a simple Lie group with equal rank; namely, the rank of $G$ equals the rank of its maximal compact subgroup. Suppose that $G'$ is a reductive subgroup of $G$ with equal rank. If $\...
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Two basic question on parabolic induction

I want to ask some basic two questions on the parabolic induction. Let $F$ be a local fields. Let $\chi_1,\chi_2$ be two characters of $GL_1(F)$ and $GL_1 \times GL_1$ be the Levi part of the ...
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Testing whether a quiver algebra is cellular with a computer

With a friend I made a program in the GAP-package QPA to check whether a given finite dimensional quiver algebra is quasi-hereditary. It is very slow since it has to go through all permutations of ...
Mare's user avatar
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Some basic question on the parabolic induction

I would like to ask some basic question about parabolic induction. Let $F$ be a local field and $G=GL_n(F)$ and $P=MN$ its parabolic subgroup whose Levis subgroup $M=GL_{n_1}(F) \times GL_{n_2}(F)$ ...
Monty's user avatar
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Higher traces in Hochschild cohomology

Let $A$ be an associative algebra over a field $k$. Let $\rho:A \rightarrow \mathrm{End}(M)$ a left module, finite dimensional over $k$. Then the map $a \mapsto \mathrm{tr}_M \rho(a)$ is a well ...
Reimundo Heluani's user avatar
5 votes
3 answers
807 views

Weyl's Branching Rule for $SU(N)$-Setting

On the Wikipedia page for restricted representations https://en.wikipedia.org/wiki/Restricted_representation there is presented a number of explicit "branching rules". In particular, there is the ...
Nadia SUSY's user avatar
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Integral structures via lattices

I am looking at the paper "p-adic Groups" by Bruhat (in the Boulder Proceedings, 1965). I have a question about one of the statements. Let $k$ be the quotient field of a complete discrete valuation ...
Mehta's user avatar
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semisimple support of character sheaves

So the essential question is: How should we think about, or if possible compute, the semisimple support of a cuspidal character sheaf? For example, let $G=SL_2$. We have the cuspidal character ...
Cheng-Chiang Tsai's user avatar
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Conjugacy class representatives for the automorphism group of a finite abelian group

Given a finite abelian group $A$, I'd like a list of conjugacy class representatives for its automorphism group ${\rm Aut}(A)$. In fact, it's not important that I have exactly one representative from ...
Matt Ollis's user avatar
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Good range and fair range

Let $G$ be a noncompact simple Lie group with complexified Lie algebra $\mathfrak{g}$. Fix a Cartan involution $\theta$, which defines a maximal compact subgroup $K$ of $G$. Take a $\theta$-stable ...
Hebe's user avatar
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Bernstein-Zelevinsky classification: viewing a representation as a subrepresentation or a quotient

$\DeclareMathOperator{\GL}{GL}$ $\DeclareMathOperator{\Ind}{Ind}$I have a question on the details of the Bernstein-Zelevinsky classification. This classification allows us to obtain irreducible ...
D_S's user avatar
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Minimum dimension of faithful representation of mapping class groups?

Let $\Sigma_{g}$ be a closed orientable surface of genus $g$. Let $d_g$ denote the minimum dimension of a faithful representation of the mapping class group of $\Sigma_g$. For $g=1$, the mapping class ...
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A robust version of Schur's lemma?

Does a robust version of Schur's lemma exist? Specifically, I was wondering about something like this: Let $B$ be a bounded operator over a vector space $V$, with underlying field $\mathbb{C}$ and ...
dimquasar's user avatar
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Modular $S$-matrix for an extended affine Lie algebra

This is a refinement of this old question of mine. In order to find an answer, I've been working my way through q-alg/9511026, which contains all the information I need. In this paper, the authors ...
AccidentalFourierTransform's user avatar
6 votes
1 answer
315 views

Irreducibility of the unramified principal series

Let $G = \operatorname{GL}_n(F)$ with the usual Borel subgroup $P = TU$. Let $\chi = \chi_1 \otimes \cdots \otimes \chi_n$ be an unramified character of $T$. Suppose that $\chi$ is regular, which is ...
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Is $G$ non-solvable?

Let $G$ be a finite group of order $2^7\cdot3^3\cdot5^2\cdot7$. Let $\mathrm{Irr}(G)$ be the set of all the irreducible $\mathbb{C}$-characters. Suppose that (1) there is a character $\chi\in\mathrm{...
C. Simon's user avatar
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A class function defined using Frobenius-Schur indicators

Let ${\rm Irr}(G)$ be the set of complex irreducible characters of a finite group $G$. The Frobenius-Schur indicator of $\chi\in{\rm Irr}(G)$ is defined to be $\epsilon(\chi):=\frac{1}{|G|}\sum_{g\in ...
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Character degrees of a finite group?

Let $G$ be a finite group, $R(G)$ be the solvable radical of $G$, and $G/R(G)$ be an almost simple group with socle $S\cong PSL_2(3^p)$, where $p$ is an odd prime. Also let $[G/R(G):S]=p$ and $\theta$ ...
asad's user avatar
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Conjectural nonvanishing of some combinatorial sums (6j symbols)

From various considerations and with the help of J. Van der Jeugt, I was led to conjecture the following property of a class of Wigner 6j-symbols: for any integers $k,m$ with $m\ge k\ge 2$, $$ \left\{...
Abdelmalek Abdesselam's user avatar
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Index of subgroup generated by characters induced from $p$-elementary subgroups in the ring of virtual characters

I posted this over on MSE, but received absolutely no love. So maybe I’ll have better luck here. It seems like a relatively easy group theory question that I’m just not seeing! It’s on the essential ...
Nicholas Camacho's user avatar
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1 answer
146 views

Descending chain property for compact Lie goups

I'm searching for a good reference that prove the descending chain propriety for compact Lie groups (i.e. every sequence $K_1\supset K_2\supset...$ of closed subgroups $K_i$ of $G$ is eventually ...
user 123935's user avatar
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A normalized embedding $\mathbb C \rightarrow \mathfrak a_M^{\ast}$ via $\tilde{\alpha}$

Let $G$ be a connected, reductive group over a field $k$. Let $S$ be a maximal $k$-split torus of $G$ with Weyl group $W$, $\Delta$ a set of simple roots of $S$ in $G$, and $P = MN$ a maximal ...
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Is being a Frobenius algebra a rare condition for local algebras?

Let $U_{r,l,q}$ be the set of finite dimensional local algebras $A$ over a finite field with $q$ elements such that $J/J^2$ is $r$-dimensional for a number $r \geq 2$ and such that $J^l=0$ for the ...
Mare's user avatar
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Plucker coordinates of flag varieties

I am interested in understanding Lemma A.2 in the paper "Moduli spaces of principal F-bundles" by varshavsky which you can find here. It uses so called "Plücker" coordinates of the flag variety for ...
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334 views

Koszul duality and coherent sheaves on projective space

There are different descriptions of the category $Coh(\mathbf{P}^{n})$. One can either describe it as modules over Beilinson's quiver algebra (Let us denote it by $A$) using the exceptional collection ...
user105178's user avatar
3 votes
0 answers
127 views

Does $G$ have a normal abelian Sylow $2$-subgroup?

Let $G$ be a finite group. Let $|G|=2^\alpha n$ where $(2,n)=1$ and $\alpha$ is a positive integer. Suppose that $\def\cd{\operatorname{cd}} n=\max \cd(G)$, and $n^2>\frac{1}{2}|G|$, where $\cd(G)$...
C. Simon's user avatar
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0 answers
515 views

How many non-isomorphic groups share the same character table?

I have been thinking for a while about how to enumerate all finite groups. The classical way e.g. here would be to go via latin squares and then try to calculate how many of those obey associativity. ...
Raphael J.F. Berger's user avatar
4 votes
0 answers
207 views

Conjecture on tilting modules for an Auslander algebra

On page 13 of "Tilting modules for the Auslander algebra of $K(x)/x^n$" the author, Geuenich, suggests that the number ($p_{n,i}$) of isomorphism classes of modules, occurring as the $i$-th summand of ...
Tom Copeland's user avatar
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10 votes
1 answer
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How many facets does the convex hull of all the roots of a root system have?

Let $V$ be an $n$-dimensional Euclidean vector space with inner product $\langle\cdot,\cdot\rangle$ and $\Phi$ an irreducible crystallographic root system in $(V,\langle\cdot,\cdot\rangle)$. Question ...
Sam Hopkins's user avatar
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2 votes
0 answers
51 views

Relations among hyperplane mirror symmetries

Let $H(n) \subset O(n)$ be the set of mirror symmetries in $\mathbb{R}^n$ with respect to $(n - 1)$-planes containing the origin. One can see that for any $a, b, c \in H(2)$ we have $abcabc = id$, ...
Mikhail Tikhomirov's user avatar
8 votes
1 answer
200 views

Reference request: Coxeter length and irreducible characters

Let $S_n$ be the symmetric group on $\{1,2,\ldots, n\}$ and $\ell$ the Coxeter length on $S_n$. There is a well-known formula to compute this length, namely for a $\pi \in S_n$ we have $$\ell(\pi) = |\...
Dirk's user avatar
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6 votes
1 answer
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Calculation of the Schur multiplier of $\mathbb Z^2$

Consider a projective representation of $\mathbb Z^2$ with $U(1)$ coefficients. I would like to find the covering group corresponding to this representation. For this, one needs to find the ...
Naren Manjunath's user avatar
3 votes
0 answers
487 views

On Local Langlands correspondences

Both over global function fields and $p$-adic fields, we have a series of conjectures under the name of “geometric Langlands conjectures”. Over global function fields of char $p$, they are due to ...
user avatar
9 votes
1 answer
399 views

Young tableaux for exceptional Lie algebras

Irreducible representations for the $A$-series Lie algebras are labelled Young diagrams, with a basis of each given by Young tableaux. Moreover, analogues exist for the $B,C$, and $D$ series. Does ...
Nadia SUSY's user avatar
2 votes
0 answers
121 views

Modular transformation of affine characters of non-simply connected groups$.$

Consider an (untwisted) affine algebra corresponding to a compact and simply-connected Lie group $G$. Under a modular transformation, its characters transform as (cf. 9612078) $$ \chi_\mu\to\sum_{\nu\...
AccidentalFourierTransform's user avatar
6 votes
1 answer
304 views

Branching from $E(6)$ to $SO(10) \times U(1)$

In $E(6)$ inspired models of supersymmetry, the inclusion of Lie subgroups $$ SO(10) \times U(1) \hookrightarrow E_6 $$ is important object of interest. See here for my motivating example. In ...
Nadia SUSY's user avatar
15 votes
2 answers
808 views

What are the periodic Dyck paths?

I changed the thread completely so that everything is now elementary linear algebra. A Dyck path of length $n$ is a list of positive integers $[c_1,c_2,...,c_n]$ with $c_i -1 \leq c_{i+1}$ for all $i$...
Mare's user avatar
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7 votes
1 answer
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On the structure of a finite group of order $144$

Let $G$ be a finite group of order $144$. Suppose there is an irreducible $\mathbb{C}$-character $\theta$ such that $\theta(1)=9$. QUESTION: Prove $G\cong A_4\times A_4$. By using Magma, we know ...
C. Simon's user avatar
  • 577
3 votes
0 answers
77 views

Decay of Fourier coefficients for Hölder functions on compact Lie groups

If $f$ is a complex-valued function on a compact Lie group $G$, we have a decomposition $f = \sum_\mu f_\mu$ corresponding to the Peter-Weyl decomposition $L^2(G) = \oplus_\mu (\dim \mu) V_\mu$. For $...
Salman's user avatar
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3 votes
1 answer
177 views

Kernel of restriction for ring of functions on reductive groups

Let $H \subset G$ be an inclusion of reductive groups over an algebraically closed field $k$ of char $0$. For simplicity, let's assume that $G$ is split and $H$ contains a maximal torus for $G$. Then ...
user125639's user avatar
1 vote
0 answers
136 views

Okada character table for n=6

Order the 13 fibonacci words $v$ of length $|v|=6$ as follows: \begin{equation} \begin{array}{ll} 1 &222 \\ 2 &2211 \\ 3 &2121 \\ 4 &2112 \\ 5 &2111 \\ 6 &1221 \\ 7 &1212 \...
Jeanne Scott's user avatar
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1 vote
1 answer
159 views

Abstracting the properties of the category $\frak{g}$-modules

Given a semisimple Lie algebra $\frak{g}$ over $\mathbb{C}$, and a finite dimensional irreducible representation $V$, with dual representation $V^*$, we know that the decomposition of $V \otimes V^*$ ...
Max Schattman's user avatar
9 votes
1 answer
822 views

Reference request: tensor induction

While working on a problem, I constructed something which looked like an induced representation, but with a tensor product instead of a direct sum. Here is a special case. Let $G$ be a group, with ...
D_S's user avatar
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8 votes
1 answer
388 views

Is a $G$-invariant character $\theta$ of $H$ extendible to $G$?

Let $G/H\cong PSL(2,11)$, and $\theta$ be an irreducible $\mathbb{C}$-character of $H$. Suppose $\theta$ is invariant in $G$ and $\theta(1)=9$. Question: Is $\theta$ extendible to $G$?
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