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

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34
votes
0answers
604 views

Class function counting solutions of equation in finite group: when is it a virtual character?

Let $w=w(x_1,\dots,x_n)$ be a word in a free group of rank $n$. Let $G$ be a finite group. Then we may define a class function $f=f_w$ of $G$ by $$ f_w(g) = |\{ (x_1,\dots, x_n)\in G^n\mid w(x_1,\dots,...
27
votes
0answers
868 views

Why are there so few quaternionic representations of simple groups?

Having spent many hours looking through the Atlas of Finite Simple Groups while in Grad school, I recall being rather intrigued by the fact that among the sporadic groups, only one (McLaughlin as I ...
20
votes
0answers
570 views

Combinatorics of Quantum Schubert Polynomials

Let $S_n$ be the symmetric group. Let $s_i$ denote the adjacent transposition $(i \ i+1)$. For any permutation $w\in S_n$, an expression $w=s_{i_1}s_{i_2}\cdots s_{i_p}$ of minimal possible length is ...
20
votes
0answers
607 views

Cauchy matrices with elementary symmetric polynomials

$\newcommand{\vx}{\mathbf{x}}$ Let $e_k(\vx)$ denote the elementary symmetric polynomial, defined for $k=0,1,\ldots,n$ over a vector $\vx=(x_1,\ldots,x_n)$ by \begin{equation*} e_k(\vx) := \sum_{1 \...
19
votes
0answers
348 views

A cohomology class associated with a complex representation of a group

$\newcommand\CC{\mathbb C}\newcommand\ZZ{\mathbb Z}\newcommand\ad{\mathsf{ad}}\newcommand\Ext{\operatorname{Ext}}$ Suppose that $G$ is a finite group and that it acts on a finite dimensional complex ...
18
votes
0answers
755 views

local equivalence of loop group representations

Let $G$ be a compact, simple, connected, simply connected (cscsc) Lie group, and let its smooth loop group $LG:=C^\infty(S^1,G)$. Given an interval $I\subset S^1$, we have the local loop group $$ L_IG ...
16
votes
0answers
268 views

What are the indecomposable $U_q\mathfrak{sl}(2)$-modules?

Let $\mathfrak g=\mathfrak{sl}(2)$. Let $\zeta$ be a primitive root of unity of even order. Say $\zeta=e^{2\pi i/6}$, for concreteness. Let $U_q\mathfrak g$ be Lusztig's integral form of the ...
16
votes
0answers
385 views

Applications of the surjectivity of Brauer's decomposition map over arbitrary fields?

Recently I've been going over some of Serre's reformulation of Brauer theory with a student, following the influential treatment in Part III of Serre's lectures (revised 1971 French edition) later ...
15
votes
0answers
341 views

Quasi-classical limit of representation theory

I am looking for a good reference on a general phenomenon of quasi-classical limit in representation theory, which relates "large" representations to measures on (co-adjoint orbits of) the associated ...
15
votes
0answers
597 views

Should the Dynkin diagrams of types $A_1$ and $B_2$ be labelled $C_1$ and $C_2$?

The labels $A$--$G$ attached to connected Dynkin diagrams are of course arbitrary, the result of historical accidents. In order to avoid repetitions, the four infinite families $A_\ell, B_\ell, C_\...
15
votes
0answers
433 views

Division fields of abelian varieties over function fields

Let $k$ be a finitely generated field (for example a finite field or a number field) and $K/k$ a finitely generated regular extension with $trdeg(K/k)=1$. Let $A/K$ be a principally polarized abelian ...
14
votes
0answers
325 views

Does the category of G-spectra know G?

I was recently in the situation of having access to the category of $G$-modules (for some group $G$ which I had forgotten), as just a category, i.e. no monoidal structure, together with the forgetful ...
13
votes
0answers
600 views

Capelli determinant = Duflo ( determinant) - was it known ?

Question briefly. Was this fact known: Capelli determinant = Duflo (determinant) ? (This is an equality of the two central elements in universal enveloping of Lie algebra $gl_n$). I googled a lot ...
13
votes
0answers
356 views

Nilpotent pro-$p$ groups

Is it true that every finitely generated (topologically) torsion free nilpotent pro-$p$ group is isomorphic to a subgroup of $U_d(\mathbb{Z}_p)$, the group of $d\times d$-upper triangular matrices ...
13
votes
0answers
477 views

Comparing the Kazhdan-Lusztig and Steinberg pre-orders

Both Kazhdan-Lusztig and Steinberg have defined pairs of preorders on $S_n$. Kazhdan and Lusztig's preorders come from their basis: We write $x\leq_L y$ if any left ideal spanned by K-L basis ...
13
votes
0answers
511 views

Noether-Deuring for injections and surjections?

Noether-Deuring theorem (not in the strongest form, but in the one I usually need): Let $L\diagup K$ be a field extension. Let $A$ be a $K$-algebra which is finite-dimensional as a vector space over $...
13
votes
0answers
725 views

representation theoretic interpretation of Jack polynomials

Monomial symmetric polynomials on $n$ variables $x_1, \ldots x_n$ form a natural basis of the space $\mathcal{S}_n$ of symmetric polynomials on $n$ variables and are defined by additive symmetrization ...
13
votes
0answers
727 views

Diagonalizing some matrices arising from Fourier transform on $S_n$.

Consider the function $f$ on $S_n$ which equals $1/n$ on all adjacent transpositions $(i,i+1)$, where we let $n+1 = 1$, and $0$ otherwise, and its Fourier transform $\hat{f}(\rho)$ evaluated at the ...
13
votes
0answers
455 views

Where do stable Kronecker coefficients live “in nature”?

Background: For a partition $\lambda$, let $\lambda[N] = (N - |\lambda|, \lambda_1, \lambda_2, \lambda_3, \dots)$, also let $\chi_\lambda$ be the corresponding irreducible character of the symmetric ...
12
votes
0answers
198 views

Is $F_{f, c, \ell}$ a $G$-harmonic polynomial?

Let $G \subset \text{GL}_n(\mathbb{C})$ be a finite subgroup. The group $G$ acts naturally on $\mathbb{C}^1[\mathbb{C}^n]$ the space of degree $1$ homogeneous polynomials in $x_1, \dots, x_n$, i..e, ...
12
votes
0answers
415 views

The derived category of integral representations of a Dynkin quiver.

Let $Q$ be a Dynkin quiver. Let $\mathbb CQ$ be its complex path algebra. It is defined in a way such that modules over $\mathbb CQ$ are the same as representations of the quiver $Q$. Let's write $\...
12
votes
0answers
232 views

Is there a common framework for Tannaka and Gabriel-Ulmer reconstruction theorems?

Gabriel-Ulmer duality is a biequivalence between the 2-category of finite limit categories and the 2-category of locally finitely presentable categories. It allows for the reconstruction of a theory ...
12
votes
0answers
378 views

When does a representation admit a spin structure?

Let $G$ be a finite group, and let $V$ be an $n$-dimensional real representation of $G$. Think of $V$ as given by a homomorphism $$ \rho_V\colon G\to O(n).$$ Write $\chi_V$ for the character of $V$. ...
12
votes
0answers
460 views

Are the extra vertices in Nakajima's doubling of a quiver related to Langlands duality?

To define a Nakajima quiver variety associated to a quiver $Q = (Q_0,Q_1)$ (vertices and arrows), one first doubles it to $Q^\heartsuit$ by attaching an extra vertex to every old vertex in $Q_0$. Then ...
12
votes
0answers
698 views

What can be said about Schur indices, given only the character table?

Let $\chi$ be an irreducible (complex) character of a finite group, $G$. The Schur index $m_{K}(\chi)$ of $\chi$ over the field $K$ is the smallest positive integer $m$ such that $m\chi$ is afforded ...
11
votes
0answers
504 views

Source of a formula for tensor product multiplicities?

This is a follow-up to a recent question by Allen Knutson here, involving a special type of tensor product multiplicity for a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ (or other ...
11
votes
0answers
569 views

Definition of a uniformly bounded dual of a group

The unitary dual of a group $G$ is the set of equivalence classes of irreducible unitary representations of $G$ with the Fell topology. (This topology is defined using convergence of positive definite ...
11
votes
0answers
361 views

Group rings isomorphic over F_p, but not over Z_p ?

Suppose given a prime $p$. Question: Do there exist finite groups $G$ and $H$ such that ${\bf F}_p G$ is isomorphic to ${\bf F}_p H$, but such that ${\bf Z}_p G$ is not isomorphic to ${\bf Z}_p H$ ? ...
11
votes
0answers
298 views

distribution of Young diagrams

Consider $\Lambda^p(C^n\otimes C^n)=\oplus_{\pi}S_{\pi}C^n\otimes S_{\pi'}C^n$ as a $GL_n\times GL_n$-module. This space has dimension $\binom {n^2}p$. I would like any information on the shapes of ...
11
votes
0answers
576 views

Class groups in dihedral extensions - some sort of Spiegelungssatz?

Let $p$ be an odd prime and let $F/\mathbb{Q}$ be a Galois extension with Galois group $D_{2p}$, let $K$ be the intermediate quadratic extension of $\mathbb{Q}$, and $L$ an intermediate degree $p$ ...
11
votes
0answers
536 views

Given an algebra, can it be realized as a block of a Hopf algebra?

During a classification problem I came across a set of algebras given as the path algebra of a quiver with relations. As an example the local ones: $k\langle x,y\rangle/x^2,y^2, xy-qyx$, where $q\in k$...
11
votes
0answers
580 views

Combinatorial identity involving the Coxeter numbers of root systems

The setup is: $R$ = irreducible (reduced) root system; $D$ = connected Dynkin diagram of $R$, with nodes numbered $1,2,...,r$; $\hat D$ = extended Dynkin diagram, nodes numbered $0,1,2,...,r$; $\...
11
votes
0answers
327 views

Is there a notion of tensor product of perfect bases of representations of Lie algebras?

Berenstein and Kazhdan define perfect bases as an "unquantized" version of crystal bases. A perfect basis is roughly a basis with a crystal structure such that $E_i\cdot v=\mathbb{C}\cdot \tilde{e}...
10
votes
0answers
443 views

Local proof of Grothendieck-Riemann-Roch theorem

There is a theorem by Feigin and Tsygan(Theorem 1.3.3 here) which they call "Riemann-Roch" theorem. Given a smooth morphism $f:S\to N$ of relative dimension $n$ and a vector bundle $E/S$ of rank $k$ ...
10
votes
0answers
203 views

$\mathcal{M}(\mathcal{D}_X)$ and $\mathcal{M}^r(\mathcal{D}_X)$ have natural tensor category structures?

Write $\mathcal{M}^\ell(\mathcal{D}_{X/S}) = \mathcal{M}(\mathcal{D}_{X/S})$ for the category of left $\mathcal{D}$-modules over $X$ and $\mathcal{M}^r(\mathcal{D}_{X/S})$ for the category of right $\...
10
votes
0answers
214 views

Irreducible representations of Weyl group of F$_4$ on zero weight spaces?

This is a follow-up to a recent question here concerning the natural representation of a Weyl group $W$ on the zero weight space of an irreducible representation $L(\lambda)$ of highest weight $\...
10
votes
0answers
185 views

Hodge Decompositions and Gamma Factors of Hasse--Weil L-Functions

Let $X$ be a projective variety over a number field $K$. If $\mathfrak{p}\unlhd\mathcal{O}_K$ is a prime ideal we can regard the Euler factor of its $m$th Hasse-Weil $L$-Function at $\mathfrak{p}$ as ...
10
votes
0answers
230 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 ...
10
votes
0answers
384 views

A question about multiplication in $G(\mathbb{C}((t)))$ and Affine Grassmannians

I am sorry to give a bounty to such a crappy question but an answer would help me a lot. I am stuck with the following simple (i guess but) technical problem. Let $G$ be a complex reductive ...
10
votes
0answers
378 views

Duality in category O vs. Duality of D-modules

Hello, I omit in the following all the words "derived, twisted, holonomic, finitely-generated...". We have the Bernstein-Beilinson equivalence between the category of $N$-equivariant $D$-modules on ...
10
votes
0answers
277 views

Can we find arbitrarily many elements of SU(2) generating a good copy of MAX($\ell_1^n$) inside VN(SU(2))?

In trying to prove that the answer to the title is "no", I was led to the following problem (which I think is equivalent to the question asked in the title, but can be stated independently). If ...
10
votes
0answers
433 views

canonical basis via Gelfand-Tsetlin basis

Do there exist explicit formulas for the action of Lusztig's canonical basis of $U_q(\mathfrak n_+)$ in the Gelfand-Tsetlin basis of a Verma module for $sl(n)$?
10
votes
0answers
873 views

Complexes of representations with complementary central charges

This is another question asking for references. There is an important phenomenon of correspondence between (complexes of) representations of infinite-dimensional Lie algebras with the complementary ...
9
votes
0answers
306 views

Why should Algebraic Geometers and Representation Theorists care about Geometric Complexity Theory?

Geometric Complexity Theory has demonstrated that Complexity Theorists should care about Algebraic Geometry and Representation Theory, but, why should Algebraic Geometers and Representation Theorists ...
9
votes
0answers
210 views

Geometric Satake and Restriction

The Geometric Satake correspondence (due to Lusztig, Ginzburg, Mirkovic-Vilonen) relates perverse sheaves on the Loop Group $\hat{G}$ (with their convolution product) to the Representations of the ...
9
votes
0answers
159 views

$A$-module is free if and only if equation involving Hilbert-Poincaré series holds, $M$ infinitely generated case

See my question here. Let $A = \oplus_{i \ge 0} A_i$ be a nonnegatively graded commutative algebra and $M$ a nonnegatively graded $A$-module. Assume in addition that $A_0 = k$ and all vector ...
9
votes
0answers
270 views

Vanishing theorems in algebraic geometry and representation theory

Garland proved vanishing theorems for the cohomology of a discrete subgroup with coefficients in a finite dimensional complex representation. As I understand it, Casselman reproved them using the ...
9
votes
0answers
118 views

Failure of surjectivity in Hotta-Springer specialization: examples for special unipotents?

Last weekend's workshop on Springer theory and its generalizations at UMass demonstrated how far the subject has expanded over four decades, but the original set-up for the Springer correspondence ...
9
votes
0answers
230 views

Calculation-free proof of the Weyl Integral formula for U(n)

The Weyl integral formula states that if f is a class function on U(n), T is the torus of diagonal matrices in U(n), and dU(n) and dT are the standard Haar measures on U(n) and T, then $$\int_{U(n)} ...
9
votes
0answers
182 views

Is an inclusion of finite groups with boolean lattice, linearly primitive?

Let $(H \subset G)$ be an inclusion of finite groups. Definition: Let $W$ be a representation of $G$, and $X$ a subspace of $W$. Let the fixed-point subspace $W^{H}:=\{w \in W \ \vert \ kw=w \ ,...