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13 votes
1 answer
1k views

Outer automorphism action on representations of $S_6$

Let $S_6$ be the symmetric group on 6 letters and let $\alpha \colon S_6 \to S_6$ be an outer automorphism (note that $S_6$ is the only permutation group that has an outer automorphism and that $\...
Sasha's user avatar
  • 39.3k
9 votes
1 answer
497 views

Highest weight representations of Kac—Moody algebras: what is inside the weight spaces?

Let $V(\lambda)$ be the unique irreducible representation of a Kac—Moody algebra $\mathfrak{g}$ with the highest weight $\lambda$. If $\mathfrak{g}$ is not of finite type, then even for $\lambda$ one ...
Andrei Smolensky's user avatar
7 votes
0 answers
597 views

Reference for shtuka and trace formula

I really want to learn the work of Laurent Lafforgue and the joint work of Zhiwei Yun and Wei Zhang. They both involve shtuka and trace formula, which I only know the basic idea. So I would like to ...
user330928's user avatar
4 votes
1 answer
615 views

Representation of Heisenberg-Weyl elements and their exponentials

There is possibly a huge literature on the subject but I am a newcomer on analytic representations and my need is rather specific. I simplify it below. Let $A,B$ be two symbols (standing for ...
Duchamp Gérard H. E.'s user avatar
6 votes
1 answer
366 views

Group of order $5p^aq^b$

In Lectures by Dan Bump on Modular representation theory, Theorem 13.14 states that whenever $G$ is a non-abelian simple group of order $|G|=p^aq^br$ for distinct primes $p$,$q$, and $r$, every $r$-...
FunctionOfX's user avatar
2 votes
1 answer
340 views

Exterior powers of $Sym^p T$ over Gr(k,n)

Let G=Gr(k,n) the Grassmannian of $k$-dimensional subspaces of $\mathbb{C}^n$ and denote by $T$ the (rank $k$) tautological bundle over $G$, and by $Sym^p T$ its $p$-th symmetric power. Is there any ...
Enrico's user avatar
  • 776
18 votes
5 answers
2k views

Good source for representation of GL(n) over finite fields?

I'd like to gain some understanding of unitary representations of GL(n) over finite fields. Any good source would be appreciated. ======== edit ========= My original question was ambiguous. ...
user1258240's user avatar
2 votes
0 answers
72 views

"Spectral gaps" of commutativity measure

There's a notion of commutativity measure $P(G)$ of a finite group $G$ which is probably folklore: count commuting pairs in $G \times G$ and divide by $|G \times G|$. There are some results: $P(...
Denis T's user avatar
  • 4,600
3 votes
0 answers
184 views

Mackey Obstruction Class with Integral Coefficients

Consider an exact sequence of groups \begin{equation} 1\rightarrow H\rightarrow K\rightarrow G \rightarrow1~. \end{equation} Mackey theory enables us to understand representations of $K$ in terms of ...
Clay Cordova's user avatar
  • 2,087
5 votes
0 answers
213 views

If an irreducible admissible representation is generic, so is its contragredient?

Let $G$ be a $p$-adic reductive group, and $\pi$ be an irreducible admissible representation of $G$ that is generic, do we know that the contragredient representation of $\pi$ is also generic? If $G$ ...
Q-Zh's user avatar
  • 960
8 votes
0 answers
134 views

Rational homotopy type of Hilbert scheme components

What is known about rational homotopy type of irreducible component of $Hilb^n(\mathbb C^k)$ containing configuration space? I've searched arXiv for a while and found nothing but surmise that Betti ...
Denis T's user avatar
  • 4,600
4 votes
0 answers
76 views

Comparing parametrizations of unipotent radical

Let ${G}$ be a simple algebraic group over $\mathbb{C}$ with maximal torus $T$ and set of simple roots $\{\alpha_i\}_{i\in \Delta}$. We then have a Borel supgroup $B=TU$ with unipotent radical $U$. ...
Spencer Leslie's user avatar
5 votes
1 answer
1k views

The Casimir invariant of an irreducible representation of a compact Lie group

Let $G$ be a compact Lie group (not necessarily connected) and $\rho:G\to \mathrm{End}(V)$ an irreducible (hence finite-dimensional) unitary representation of $G$. Let $\mathfrak{g}$ be the Lie ...
B K's user avatar
  • 1,942
4 votes
2 answers
288 views

The explicit indecomposable representations of (any) Euclidean quiver of type E

It is known that for any quiver $Q$ that is an orientation of $\tilde{\mathbb{E}}_8$, the hereditary path algebra $KQ$ ($K$ being an algebraically closed field) is tame (but not finite). That is, in ...
Iteraf's user avatar
  • 482
6 votes
0 answers
1k views

Definition of Admissible Representation

Let $G$ be a connected, reductive group over a number field $k$. Let $v$ be a place of $k$. If $v$ is finite, an admissible representation of $G(k_v)$ is defined to be an abstract representation of $...
D_S's user avatar
  • 6,180
2 votes
0 answers
571 views

Clebsch–Gordan(CG) coefficients for SO(N) and Sp(N) group

I know how to calculate the CG coefficients for $SU(N)$, but there are other simple Lie group like $SO(N)$ and $Sp(N)$. But up to now I can't find any textbook tells me how to calculate these and I ...
fff123123's user avatar
  • 249
2 votes
0 answers
169 views

integrable modules reference request

Let $\mathfrak{g} = \displaystyle\varinjlim_{n} \mathfrak{g}_n$ be a locally simple Lie algebra. Let $M$ be an integrable module over $\mathfrak{g}$, that is, for every $n$, as a $\mathfrak{g}_n$ - ...
Alex's user avatar
  • 501
6 votes
1 answer
242 views

Imbedding of a representation of a compact subgroup

Let $G$ be a compact subgroup of $O(n)$. Let $\rho$ be a continuous finite dimensional representation of $G$. Question Is it true that there exists a continuous finite dimensional representation $\...
asv's user avatar
  • 21.8k
4 votes
2 answers
829 views

Reference Request: Definition of Induced Representation for reductive groups over a local field

Let $G$ be a connected, reductive group over a local field $F$ of characteristic zero, and $H$ a closed subgroup of $G$ which is defined over $F$. Let $\mu_H, \mu_G$ be right Haar measures on $H(F), ...
D_S's user avatar
  • 6,180
6 votes
1 answer
148 views

Equivalent orthogonal representations are orthogonaly equivalent

Could anyone point me to a reference for the following fact : Let $G$ denote the orthogonal or symplectic complex group and $H$ be a complex Lie group, then if $\rho_1\,:H\, \rightarrow G$, $\rho_2\,...
JadeSnail's user avatar
  • 474
11 votes
0 answers
818 views

How to compute Ext-groups for categories without enough injectives/projectives?

I am studying a category of representations of an infinite-dimensional Lie algebra, which is an abelian category. Unfortunately, the category does not have enough injectives/projectives. I would ...
Batominovski's user avatar
11 votes
1 answer
620 views

Gauss, Jacobi, Kloosterman sums and representation theory in the $\mathbb F_1$-world

This question is inspired by Why are Bessel function and Kloosterman sum similar? - it developed in me desire to understand Kloosterman sums better. There seems to be common knowledge that Gauss, ...
მამუკა ჯიბლაძე's user avatar
5 votes
0 answers
303 views

Recovering an A-infinity structure on an Ext-algebra from a quiver presentation

Let $A=KQ/I$ be a basic finite dimensional algebra given by a quiver with relations. Let $S$ denote the direct sum of the corresponding simple modules. According to [Keller: A-infinity algebras in ...
Julian Kuelshammer's user avatar
3 votes
1 answer
225 views

$f^{\lambda}$: asymptotics and analytic continuations

Let $\mathbb{Y}_n$ denote the set of all partitions of $n\in\mathbb{N}$ and $\mathbb{Y}$ Young's lattice of all partitions. The partition function $g_0(n)=\sum_{\lambda\in\mathbb{Y}_n}1$ has an ...
T. Amdeberhan's user avatar
7 votes
3 answers
1k views

the character tables of irreducible representations of $SL(3,Z_q)$

The following paper gives a classification of the character tables of irreducible representations of $SL(3,GF(q))$ where $q$ is a power of a prime number, and $ GF(q)$ a finite field of $q$ elements. ...
Xiao-Gang Wen's user avatar
3 votes
0 answers
116 views

Extension of representations of certain compact Lie groups

Let $G$ be a real, connected, non-compact, semisimple Lie group with finite center and real rank $1$. Let $G=KAN$ be an Iwasawa decomposition, then $K\subset G$ is a maximal compact subgroup, and $\...
B K's user avatar
  • 1,942
2 votes
1 answer
431 views

Lagrange interpolation vs homogeneous symmetric polynomials?

This question is a follow-up on another MO query here. Question. For $r\geq$ an integer, is it true that there exists homogeneous symmetric polynomial $P_r(x_1,\dots,x_n)$ with positive ...
T. Amdeberhan's user avatar
3 votes
1 answer
372 views

Reference for real and complex projective representation of finite group

I'm not a mathematician. I've only learnt about irreducible representation of finite group, symmetric group and simple Lie group. In fact, I don't know projective representation belong to which part ...
346699's user avatar
  • 977
4 votes
1 answer
628 views

Closed orbits for the action of general linear groups

Let $G=GL_n(K)$ where $K$ is an algebraically closed field of characteristic zero. Let $V$ be a finite dimensional rational representation of $V$. Assume that $v\in V$ has a reductive stabilizer $H\...
Ehud Meir's user avatar
  • 5,039
5 votes
1 answer
250 views

About generalized Springer theory for spin groups

I am interested in the detailed computation of the generalized Springer theory for spin groups (type B or D). In the last sentence in Section 14 of Lusztig's Intersection cohomology complex on a ...
Cheng-Chiang Tsai's user avatar
1 vote
0 answers
75 views

Representation-finite trivial extension algebras

Given a quiver algebra A such that its trivial extension T(A) is representation-finite. Is T(A) automatically stable equivalent to a trivial extension algebra of a hereditary representation-finite ...
Mare's user avatar
  • 26.5k
9 votes
0 answers
409 views

The proof of Kazhdan's density theorem (And does it hold over positive characteristic?)

When proving identities about traces of functions on representations of $p$-adic groups, Kazhdan's density theorem indicates one only has to check equalities of traces on tempered representations. ...
Charles Denis's user avatar
5 votes
1 answer
261 views

A "prequestion" about meromorphic representations of algebraic groups

In a comment exchange around an answer to Is a group scheme determined by its category of representations? there arose the issue of Tannakian reconstruction for non-affine algebraic groups (e. g. ...
მამუკა ჯიბლაძე's user avatar
2 votes
1 answer
75 views

Contravariant finiteness of subcategories

Let $A$ be a finite dimensional algebra . Let $P_{\inf}$ be the full subcategory of modules having finite projective dimension and $P_r$ the subcategory of modules having projective dimension bounded ...
Mare's user avatar
  • 26.5k
3 votes
0 answers
205 views

Finitistic dimension via tilting modules

is the following true (all algebras and modules are assumed to be finite dimensional): The finitistic dimension of an algebra is equal to the supremum of projective dimensions of tilting modules? It ...
Mare's user avatar
  • 26.5k
1 vote
0 answers
136 views

Representations of finite groups over commutative rings-question and reference request

In a textbook of representation theory I have encountered the following statement without proof: Let $R$ be a commutative ring and $G$ a finite group. If $M$ is a simple $RG$-module then the ...
user103474's user avatar
6 votes
0 answers
184 views

Reference request: fusion rules for unitary dual of SL(3,R)?

By the fusion rules, I mean: given two unitary irreps of the group, what unitary irreps occur in their tensor product and with what "multiplicity"? (I am guessing that direct integrals ...
Yemon Choi's user avatar
  • 25.8k
6 votes
0 answers
233 views

Tracking down a copy of "Mixed categories, Ext-duality and representations (results and conjectures)"

I am trying to find a copy, ideally digital, of the following preprint: Title: Mixed categories, Ext-duality and representations (results and conjectures) Authors: A. Beilinson and V. Ginzburg Year:...
uncookedfalcon's user avatar
0 votes
1 answer
337 views

Homological dimensions of tensor products of algebras

Given two finite dimensional algebras $A$ and $B$ over a field. The Gorenstein dimension of an algebra A is defined as the injective dimension of the module A. The finitistic dimension of an algebra A ...
Mare's user avatar
  • 26.5k
1 vote
0 answers
98 views

Ireducible representations in characteristic p [duplicate]

It is known that the intersection of kernels of all ireducible representations of a finite group in characteristic zero is the trivial group. In characteristic $p>0$ I have understood that this ...
mathuser17's user avatar
7 votes
1 answer
168 views

Region of convergence of Eisenstein series is a union of Weyl chambers when groups have discrete series?

Let $G$ be a reductive algebraic group, and let $M$ be a Levi subgroup of $G$. In Urban's paper Eigenvarieties for Reductive Groups, it seems to be assumed that if $G(\mathbb{R})$ and $M(\mathbb{R})$ ...
dgulotta's user avatar
  • 913
8 votes
1 answer
476 views

How does Jacquet's "Generic Representations" classify tempered representations?

Let $L$ be a $p$-adic field $G = GL_n(L)$. Let $P$ be a standard parabolic subgroup with Levi decomposition $P = MU$, where $M \cong G_1\times \ldots \times G_r$, for $G_i \cong GL_{n_i}(L)$. The ...
John Binder's user avatar
  • 1,453
15 votes
2 answers
762 views

Gelfand-Tsetlin algebras and "Jucys-Murphy elements" for $\mathfrak{gl}_n$

I'm trying to figure out/find in literature the details concerning Gelfand-Tsetlin algebras for $\mathfrak{gl}_n(\mathbb C)$ (Okounkov-Vershik style, if you wish). Consider the chain $$\mathcal U(\...
Igor Makhlin's user avatar
  • 3,513
25 votes
3 answers
1k views

what else is in $\prod_{j=1}^n(1+q^j)$?

From time to time, I run into the finite product $\prod_{j=1}^n(1+q^j)$. And, the more it happens, the more fascinated I've become. So, herein, I wish to get help in collecting such results. To give ...
T. Amdeberhan's user avatar
6 votes
0 answers
268 views

duality between quiver variety and affine Grassmannian

Let $\frak{g}$ be a ADE type simple lie algebra. There are (at least) two geometric ways to get highest weight irreducible representations of $\frak{g}$. One is by considering constructible functions ...
Ben's user avatar
  • 849
7 votes
1 answer
222 views

Some intuition on the $SL_n$-module $V_{[1,1,...,1]}$

(This question highly overlaps with this and also this.) The irreducible ${\sf SL}_{n-1}$-module $V_{[1,1,\ldots,1]}$ is the one providing the minimal projective embedding $\mathbb{P}(V_{[1,1,\ldots,...
Giovanni Moreno's user avatar
5 votes
1 answer
242 views

Jack polynomials and the Witt algebra

The symmetric Jack polynomials $J_n^{\alpha}(x_1,x_2,..,x_{n+1})$, a special subset of the symmetric Jack functions presented in Stanley's paper in equation a) on page 80, can be represented by the ...
Tom Copeland's user avatar
  • 10.5k
11 votes
1 answer
626 views

Formula for $U(N)$ integration wanted

Before you jump on the "duplicate" buttom, let me say that I do not want to hear about Weingarten calculus and I do not want to see a character of the symmetric group. What I would like is a formula ...
Abdelmalek Abdesselam's user avatar
4 votes
1 answer
345 views

Irreducible representations of the reductive quotient

For a linear algebraic group over an algebraically closed field of characteristic zero $G$, with unipotent radical $U$, we have that $G/U$ is reductive. When $G$ is solvable, then Lie's theorem says ...
math no more's user avatar
  • 1,423
6 votes
2 answers
2k views

Alternative or reprint of Carter's "Finite Groups of Lie Type: Conjugacy Classes and Complex Characters"

I would like to learn about character theory of finite groups of Lie type and some Deligne-Lusztig theory. The classic textbook on the subject seems to be Roger W. Carter's Finite Groups of Lie Type: ...
Gro-Tsen's user avatar
  • 32.5k

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