# Questions tagged [rt.representation-theory]

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

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### Why do combinatorial abstractions of geometric objects behave so well?

This question is inspired by a talk of June Huh from the recent "Current Developments in Mathematics" conference: http://www.math.harvard.edu/cdm/.
Here are two examples of the kind of combinatorial ...

**44**

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782 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,...

**32**

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417 views

### Chern character of a Representation

Let $G$ be a finite group. Under the identification of the representation ring $R_{\mathbb{C}}(G)$ with the equivariant K-theory $KU^0_G(\ast)$ of the point, followed by Atiyah-Segal completion-...

**32**

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1k 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 ...

**29**

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462 views

### Groups whose complex irreducible representations are finite dimensional

By a complex irreducible representation of a group $G$, I mean a simple $\mathbb CG$-module. So my representations need not be unitary and we are working in the purely algebraic setting.
It is easy ...

**22**

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890 views

### Linking formulas by Euler, Pólya, Nekrasov-Okounkov

Consider the formal product
$$F(t,x,z):=\prod_{j=0}^{\infty}(1-tx^j)^{z-1}.$$
(a) If $z=2$ then on the one hand we get Euler's
$$F(t,x,2)=\sum_{n\geq0}\frac{(-1)^nx^{\binom{n}2}}{(x;x)_n}t^n,$$
on the ...

**22**

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707 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 ...

**21**

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516 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 $V$.
...

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391 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 ...

**20**

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776 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**

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299 views

### “quantum” symmetric plane partitions beget alternating sign matrices?

The "quantum" version qTSPP of the number of totally symmetric plane partitions, contained in the cube $[0,n]^3$, is enumerated by
$$f_n(q):=\prod_{j=1}^n\prod_{k=1}^j\prod_{\ell=1}^k\frac{1-q^{j+k+\...

**18**

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458 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 ...

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444 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 ...

**18**

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833 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 ...

**18**

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690 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_\...

**17**

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415 views

### Who first noticed the duality for finite groups?

A.A.Kirillov in section 12.3 of his "Elements of the Theory of Representations" writes that the first "symmetric" duality theory for non-commutative groups was the theory for finite groups. In short ...

**17**

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708 views

### McKay conjecture for finite groups in the simplest case G=GL(2,F_p) ( puzzle: Borel knows about cuspidals)

The McKay conjecture and related (Alperin, Issacs-Navarro) are one of the "main problems in the representation theory of finite groups" (G.Navarro pdf).
Statement of the McKay conjecture is quite ...

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941 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 ...

**17**

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742 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 $0,1,2,...,r$;
$\...

**16**

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508 views

### When is the determinant an $8$-th power?

I am working over $\mathbb{R}$ (though most of the story goes over any field). I am looking for linear spaces of matrices such that the restriction of the determinant to this spaces can be written (...

**16**

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536 views

### Monstrous Langlands-McKay or what is bijection between conjugacy classes and irreducible representation for sporadic simple groups?

Context: The number of conjugacy classes equals to the number of irreducuble representations (over C) for any finite group. Moreover for the symmetric group and some other groups there is "good ...

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856 views

### What to do with results you found but cannot prove(outside your research area)?

Not sure if MathOverflow is still a place to discuss such things, but I'll give it a try. Tell me an alternative site, in case it is wrong here. I translated a representation-theory/combinatorial ...

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528 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 ...

**16**

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876 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 ...

**15**

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413 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 ...

**14**

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247 views

### The Monster Moonshine Module from the engineering or algorithmic point of view

From what I understand (see, e.g., this question), the Monster Moonshine Module is a kind if "third generation" (or "second quantization"?) after the Golay code (with automorphism group $M_{24}$) and ...

**14**

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247 views

### Representation categories and homology

Let $G$ be a finite group.
Let $\mathcal{C}=Rep-G$ be the rigid $\mathbb{C}$-linear symmetric monoidal category of finite dimensional complex representations of $G$.
Can we recover some homological ...

**14**

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379 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 ...

**14**

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712 views

### Dual versions of “folding” symmetric ADE Dynkin diagrams?

Start with the Dynkin diagram of an irreducible root system, typically associated with a simple
Lie algebra over $\mathbb{C}$ or a simple algebraic group. Most of the simply-laced ADE
diagrams ...

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672 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 ...

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496 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**

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203 views

### Group homology $\mathrm{SL}_2$ acting on $\mathrm{Sym}^g$

Let $k$ be a field. We write $\mathrm{Sym}^g(k^2)$ for the $g$-th symmetric power of the (a?) standard representation of $\mathrm{GL}_2(k)$ ($g\geq 0$ an integer). Here I consider $\mathrm{Sym}^g(k^2)$...

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263 views

### Analog of Haar element in an algebra

In a Hopf algebra $H $ (over some field $ k $), there is the notion of a Haar element $ h \in H$. This is an element of the algebra which has the property that if $ V $ is a representation of $ H $, ...

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540 views

### Special configurations on a circle from a homological algebra problem

Here is the short version of the combinatorial problem:
Given a positive integer $n \geq 2$. Draw a circle with $2n$ points indexed by the numbers from $\mathbb{Z}/ 2n \mathbb{Z}$. We colour the ...

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597 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 ...

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705 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 $k$ ...

**13**

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710 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:...

**13**

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457 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 ...

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827 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 ...

**12**

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503 views

### Representation theory of finite groups with additional structures

Let $H$ be a finite group, representation theory of $H$ over $\Bbb C$ essentially determines $\operatorname{Hom}(H,GL_n(\Bbb C))$ up to conjugation action of $GL_n(\Bbb C)$ for each $n$. If we replace ...

**12**

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235 views

### Has Kac's conjecture (*), from “Infinite root systems, representations of graphs and invariant theory”, been proved?

Let $k$ be an algebraically closed field of characteristic zero, $V$ a finite dimensional $k$ vector space, $V^{\ast}$ the dual space, and $G$ an algebraic subgroup of $GL(V)$. Let $V_0$ be the points ...

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465 views

### Converse of Frobenius

Enumerate the elements of a finite group $G$ as follows: $g_1,g_2,\dots,g_n$. Introduce $n$ variables indexed by the elements of $G$: $x_{g_1},\dots,x_{g_n}$.
Consider the matrix $X_G$ with entries $...

**12**

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567 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 ...

**12**

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388 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 ${\bf Z}_p H$ ?
...

**12**

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504 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**

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286 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**

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571 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 $q\in k$...

**12**

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494 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 ...

**11**

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198 views

### 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 ...

**11**

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287 views

### 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\{...