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

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4
votes
1answer
175 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 ...
10
votes
2answers
234 views

Can one describe the multiplication of two Bruhat cells?

For $G$ a simple linear algebraic group and $B$ a fixed Borel subgroup, we have the Bruhat decomposition $G = \coprod_{w \in W} B\dot{w}B$, where $W$ is the Weyl group and $\dot{w}$ is any ...
2
votes
1answer
224 views

Is there a link between $H_2(G,\mathbb{Z})$, the Schur Multiplier of a group, and the “other” Schur multipliers of a group?

The name for the the following 2 mathematical objects: $$H_2(G,\mathbb{Z})$$ and $$\{K:G\times G\longrightarrow\mathbb{C}\ |\ \forall T\in B(l^2(G))\text{we have that}~S:G\times ...
5
votes
1answer
100 views

Reference request: normalization of intertwining operators for GL(2, C)

Take $F$ a local field and $\chi_1, \chi_2$ two characters, write $M(\chi_1, \chi_2)$ for the standard intertwining integral $$M(\chi_1. \chi_2).f(g) := \int_{F} f\left( \begin{pmatrix} 0&-1\\ ...
0
votes
0answers
66 views

Relation between quantum affine algebras and W-algebras

In the paper, there is a concept $qq$-characters for W-algebras $W_{q_1,q_2}$. The usual $q$-character is defined for quantum affine algebra in the paper. A $q$-character can be obtained from a ...
5
votes
1answer
124 views

The action of $GL_{\infty}$ on the infinite wedge space

This is a question from the book "Highest weight representations of infinite dimensional Lie algebras, 2nd ed" by V. G. Kac, A. K. Raina, and N. Rozhkovskaya. Consider the following objects: the ...
3
votes
0answers
81 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 ...
16
votes
5answers
547 views

Is there a formula for the Frobenius-Schur indicator of a rep of a Lie group?

Let $G$ be a simple algebraic group group over $\mathbb C$. Let $V$ be a self-dual representation of $G$. Let $\lambda$ be the highest weight of $V$. Write $\lambda$ as a sum of fundamental weights: ...
7
votes
1answer
248 views

Ring of invariants for the regular representation

The symmetric group $S_n$ acts on $\mathbb C^n$ by permuting the coordinates. In this case the ring of invariants is generated by elementary symmetric polynomials in n-variables. Now consider the ...
3
votes
0answers
106 views

The left regular representation of the Jacobi groups over local fields of characteristic >2 is type I?

Let $K$ be a non-archimedean local field of characteristic $>2$. Consider the Jacobi group $G=H_{2n+1}(K)\rtimes Sp_{2n}(K)$, which is the semidirect product of the Heisenberg group $H_{2n+1}(K)$ ...
0
votes
0answers
94 views

Functional composition of Hadamard product

Let $\Bbb K$ be a ring. Are there universal functions $$f,h:\Bbb K^{n\times n}\times\Bbb K^{n\times n}\times\Bbb K^{n\times n}\times\Bbb K^{n\times n}\rightarrow\Bbb K^{n\times n}$$ $$g:\Bbb ...
14
votes
2answers
475 views

factorization of the regular representation of the symmetric group

Let $\mathbb{C}[S_n]$ be the regular representation of the symmetric group $S_n$, and let $\mathbb{C}^n$ be the vector representation. Question: Does there exist a representation $V$ (of dimension ...
3
votes
0answers
105 views

Metaplectic groups over non-archimedean local fields of characteristic>2

Let $K$ be a non-archimedean local field of characteristic $>2$. Consider the double cover metaplectic extension of symplectic groups $p: Mp_{2n}(K)\rightarrow ...
36
votes
1answer
1k views

What is the status of Arthur's book?

Arthur's long-awaited book project is now published (The endoscopic classification of representations: orthogonal and symplectic groups). However, in the book he takes some things for granted: The ...
2
votes
0answers
212 views

algebraic representation over $\mathbb{C}$

In reading the Harris-Taylor book, I encounter expressions like "Let $\xi$ be an algebraic representation of $G$ over $\mathbb{C}$". What does this mean? Here $G$ is a reductive group over ...
2
votes
0answers
59 views

Determine the representation given by space of sections of symmetric products of cotangent bundle of projective plane

In a recent project, it was interesting for me to determine the $PGL(3)$ representation given by $H^0(S^2(\Omega(1)) \otimes \mathcal O(4))$ on $\mathbb P^2$. I did this by using the Euler sequence, ...
4
votes
1answer
104 views

Homological characterisation of standardly stratified algebras using Ext

Let A be a finite dimensional algebra and $S_1,S_2,...,S_n$ the simple $A$-modules and $P_1,..,P_n$ the indecomposable projective $A$-modules. For $i=1,...,n$, define the standard module $\Delta_i$ as ...
17
votes
3answers
489 views

Is there a short proof that the Kostka number $K_{\lambda \mu}$ is non-zero whenever $\lambda$ dominates $\mu$?

This is maybe a little basic for MathOverflow, but I'm hoping it will get some interesting answers. Let $\unrhd$ be the dominance order on partitions of $n \in \mathbb{N}$. For partitions $\lambda$ ...
3
votes
1answer
88 views

Why are the convolvers in the bicommutant of the pseudo-measures? ($CV_p(G)\subseteq PM_p(G)''$)

Let $G$ be a locally compact group. For $1<p<\infty$ let $\lambda_p:G\to\mathcal{B}(L^p(G))$ (resp. $\rho_p:G\to\mathcal{B}(L^p(G))$) be the left (resp. right) regular representation. $CV_p(G)$ ...
2
votes
1answer
97 views

Modularisation on group representations with arbitrary braiding

Applying the modularisation/deequivariantisation procedure to the representation category $\operatorname{Rep}_G$ of a finite group $G$ with trivial braiding gives the fibre functor to vector spaces. ...
5
votes
1answer
125 views

Is this modified bound quiver algebra necessarily representation-finite?

Suppose that $A = kQ/I$ is a bound quiver algebra for $k$ an algebraically closed field, $Q=(Q_0, Q_1)$ a finite connected quiver with no oriented cycles with no multiple edges or self-loops, and $I$ ...
2
votes
1answer
166 views

Every norm-decreasing algebra morphism $L_1(G)\to\mathcal{B}(E)$ comes from a group representation

In section 8 of this paper http://arxiv.org/abs/math/0611833v3 the author proves the following: If $E$ is a reflexive Banach space, $G$ a locally compact group and $\pi:L_1(G)\to\mathcal{B}(E)$ a ...
2
votes
1answer
76 views

Is a quotient of a bound quiver algebra of finite representation type also representation-finite?

Let $A = kQ/I$ be a bound quiver algebra for some algebraically closed field $k$, $Q$ a finite connected quiver without oriented cycles, and $I$ an admissible ideal. Say that $I'$ is also an ...
0
votes
1answer
102 views

Perfect $Q[G]$-complex

Let $G$ be a finite group and let $M$ be a perfect $\mathbb{Q}[G]$-complex. Suppose that $M\otimes_{\mathbb{Q}[G]}\mathbb{Q}$ is quasi-isomrphic to $0$ can we conclude that $M$ is quasi-isomorphic to ...
4
votes
0answers
81 views

Classification of representation-finite algebras up to stable equivalence of Morita type

assume K is an algebraically closed field. I wanted to ask if there is a classification of the representation-finite K-algebras up to stable equivalence of Morita type at least for some small numbers ...
0
votes
0answers
49 views

Normal conjugate of elements of unipotent upper tringular matrices over F_q

Let $UT_n(q)$ be the group of upper triangular matrices with entries in the finite field $F_q$ and ones on the diagonal. Denote the normal closure of an element $s\in UT_n(q)$ by $s^{UT_n(q)}$, i.e., ...
17
votes
14answers
1k views

Applications of Representation Theory in Combinatorics

What are the examples of interesting combinatorial identities (e.g. bijection between two sets of combinatorial objects) that can be proved using representation theory, or has some representation ...
0
votes
0answers
23 views

Generators for equivariant polynomial maps in the case of finite reflection groups

Let $G$ be a finite reflection group acting on a finite dimensional $\mathbb{R}$ vector space $E$. Then $Mor_G(E,E^*)$ is a free $\mathbb{R}[E]^G$ module. Moreover if $f_1,\ldots,f_n$ are the basic ...
6
votes
4answers
578 views

When does $\langle grg^{-1}|g\in G\rangle = \langle gsg^{-1}|g\in G\rangle$ imply $\langle r\rangle=\langle xsx^{-1}\rangle$?

We call a finite group $G$ normal if for all $s,r\in G$, if $\langle gsg^{-1}:g\in G\rangle =\langle grg^{-1}:g\in G\rangle $ then there exists $x\in G$ such that $\langle r\rangle =\langle ...
2
votes
1answer
133 views

A canonical representative in Morita equivalence class

Let $A$ be a finite dimensional algebra over an algebraically closed field $K$. If $A$ is semisimple, then $A$ is Morita equivalence with a commutative algebra, that is $A \backsim K^n$ where $n$ is ...
5
votes
0answers
166 views

Homogeneous spaces of affine algebraic groups

Let $G$ be a reductive algebraic group over an algebraically closed field $K$ of characteristic zero (I am particularly interested in the case $G=GL_n(K))$. Let $H$ be a closed subgroup of $G$. It is ...
7
votes
2answers
321 views

Request for classical articles in representation theory

I am planning in running a Ph.D. student seminar next year on representation theory in the spirit of MIT Kan's Seminar where students give lectures on classical articles on representation theory that ...
8
votes
1answer
183 views

$U_q(\mathfrak{sl}_2)$ representations of “quantum dimension” zero

I'm reading up on quantum groups and their applications and I've come across a question I just can't find an answer to. I know about the basic representation theory of $U_q(\mathfrak{sl}_2)$ and I ...
6
votes
0answers
73 views

Correlation of Class Functions

Let $G$ be a finite group, and let $f_1,f_2$ be two real-valued class functions of $G$. Assume that multiplying elements of $G$ takes $O(1)$-time. Let $s:G\to \mathbb{R}$ be defined by ...
4
votes
1answer
83 views

Construction of vector space isomorphism where $f(v \otimes gh) = \sigma(g)(f(v \otimes h)),\text{ }\forall v \in V,\text{ }g,\,h \in G$

Let $G$ be a finite group, $R = \mathbb{C}G$ the regular representation of $G$, and $\rho : G \to \text{GL}(V)$ a finite dimensional representation of $G$. Write $\sigma: G \to \text{GL}(V \otimes R)$ ...
5
votes
1answer
185 views

Description of the algebra of $G$-invariant polynomials by generators and relations

Fix $n > 1$ and let $\zeta \in \mathbb{C}$ be a primitive $n$-th root of unity. Let $G \subset \text{SL}_2(\mathbb{C})$ be a cyclic subgroup of order $n$ generated by the diagonal matrix $g = ...
3
votes
0answers
325 views

Mystery behind ADE Dynkin diagram [duplicate]

ADE Dynkin diagram classifies: finite type quiver (Gabriel's theorem) du Val singularity (minimal resolution of $\mathbb{C}^2/G$, $G$ is a finite subgroup of $SL_2(\mathbb{C})$) simply laced simple ...
0
votes
0answers
72 views

Orbits of some action of SL2 on Pontryagin dual of the field of formal Laurent series

Let $K=\mathbb{F}_2((t))$ be the field of formal Laurent series over the finite field $\mathbb{F}_2$. Now consider $K^3$ as an additive group and its dual group $\hat{K^3}$, which consists of all ...
1
vote
1answer
247 views

Decompose $\Lambda^3(V \otimes W)$ [closed]

Let $V, W$ be two vector spaces. We have $\Lambda^2(V \otimes W) \cong (\Lambda^2 V \otimes S^2 W) \oplus (S^2 V \otimes \Lambda^2 W)$. I am trying to find similar results for $\Lambda^3(V \otimes ...
0
votes
0answers
81 views

Is there a metric defined on the product space of orthogonal groups?

If one considers just the orthogonal group, then there is a natural metric given by [1]: $$\begin{align} \theta & =\frac{1}{2} \| \log(R_1^{-1}R_2) \| \\ & = \frac{1}{2} \| ...
3
votes
1answer
222 views

A question of the book Elements of the representation theory of associative algebras volume 1

I'm reading the book "Elements of the representation theory of associative algebras, volume 1". And I can't understand the proof of the proposition 3.11 on page 124 (the place where marked green). ...
4
votes
1answer
292 views

Quotient of a vector space by a linear finite group action

Let the cyclic group $\mathbb{Z}_n$ act on $\mathbb{C}^n$ (or on $\mathbb{R}^n$, I'm interested in both) by permuting coordinates. What does the topological quotient $Q$ by this group action look ...
2
votes
0answers
121 views

Differential of the adjoint quotient map

My question is regarding a paper by R.W Richardson titled "Derivatives of invariant polynomials on a semisimple Lie Algebra" ** . In this paper, he reports on computations of the rank of the ...
4
votes
0answers
158 views

Hochschild cohomology of SU(2)

I have a question about the computation of an Hochschild Cohomology. Or at least about a space which really looks like a cohomology space. All the functions i consider are assumed to be smooth. Let's ...
2
votes
0answers
42 views

Calculation of minimal right $\operatorname{add}(M)$-approximations

given a finite dimensional quiver algebra $A$ and a generator $M$ with $\operatorname{Ext}^1(M,M)=0$. By Wakamatsus lemma, for any $A$-module $N$ there exists a surjective $A$-linear map $f\colon M_1 ...
0
votes
0answers
79 views

What are the explicit expressions of quantum Casimir elements for $U_q(sl_3)$ and $U_q(sl_4)$?

What are the explicit expressions of quantum Casimir elements for $U_q(sl_3)$ and $U_q(sl_4)$ in terms of $E_1, E_2, F_1, F_2, K_1, K_2, K_1^{-1}, K_2^{-1}$? Any help will be greatly appreciated!
5
votes
1answer
163 views

Strongly real elements of odd order in sporadic finite simple groups

Recall that an element of a finite group is said to be real if it is conjugate to its inverse, and strongly real if the conjugating element can be chosen to be an involution. Question: Is it true ...
16
votes
1answer
441 views

Okounkov-Vershik approach to representation theory of $S_n$

This is a rather soft question. I was wondering if someone could explain on a fundamental and intuitive level, what the Okounkov-Vershik approach to representation theory of $S_n$ is all about. It's ...
8
votes
0answers
146 views

$k[x_1, \dots, x_n]$ is free iff $\mathbb{C}[x_1, \dots, x_n]^G \otimes \text{Harm}(\mathbb{R}^n, G) \to k[x_1, \dots, x_n]$ isomorphism?

For any subgroup $G \subset \text{GL}_n(\mathbb{R})$ the set $\mathbb{C}[x_1, \dots, x_n]^G$, of $G$-invariant polynomials, is a graded subalgebra of $\mathbb{C}[x_1, \dots, x_n]$, resp. the set ...
2
votes
0answers
87 views

what is the link between plethysm in regular representation of the symmetric group and plethysm in Schur functions.

I am trying to understand first how one can define the plethysm say $s_\lambda \circ s_\mu$ as a module in the regular representation of the symmetric group. 1)How is it connected to the plethysms ...