# Questions tagged [rt.representation-theory]

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

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### Openly available software to work with Demazure modules

Does someone know of any sort of software openly available online which can be used to compute various characteristics of Demazure modules for semisimple Lie algebras? Specifically, I'm interested in ...

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### Extension of Verma modules

The category $\mathcal{O}$ is the category of all finitely generated, locally $\mathfrak{b}$-finite and $\mathfrak{h}$-semisimple
$\mathfrak{g}$-modules, where $\mathfrak{g}$ is a complex semisimple ...

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### Definition of a Dirac operator

So it seems that a Dirac operator acting on spinors on $\psi=\psi(\mathfrak{su}(2),\mathbb{C}^2)$ can be written in this case simply as:
$D=\sum_{i,j} E_{ij}\otimes e_{ji}$, where $E_{ij}$ are ...

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### Proving that $\lambda\mapsto \chi^\lambda(C)/f^\lambda$ is a polynomial

Let $\lambda$ be a partition of $n$ and $\chi^\lambda$ be the character of $S_n$ associated to it. Given any conjugacy class $C$, I want to prove that
$$\lambda\mapsto \frac{\chi^\lambda(C)}{f^\lambda}...

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### Computing affine Springer fibers

I'm having some trouble computing affine Springer fibers, even in simple cases. For example, consider the group $G=SL_2$ over $\mathbb{C}$ and let $\mathcal{K}=\mathbb{C}((z))$ and $\mathcal{O}=\...

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### Schur function on unit circles

Define $T^d$ as following
$$ T^d = \left\{(t_1,\cdots,t_d)\in\mathbb{C}^{d}\mid |t_i|= 1 \mbox{ for all } i\right\}
$$
For any partition $\lambda\vdash n$,The Schur function is defined
$$
\...

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### Verma modules in category $\mathcal{O}^\mathfrak{p}$

Let $\mathfrak{g}$ be a complex semisimple Lie algebra and let $\mathfrak{h}$ be a Cartan subalgebra
of $\mathfrak{g}$. Fix a Borel subalgebra $\mathfrak{b}$ containing $\mathfrak{h}$ and fix a ...

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### Levi subgroup of Siegel parabolic of GSpin

I consider the group $G=\mathrm{GSpin(V)}$ as in this question.
We have the so called Siegel parabolic $P$ (after fixing a cocharacter) and the associated Levi $M$ (these can also be obtained using ...

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### Borel's presentation for the cohomology of a Flag Variety

If $G$ is a simple complex Lie group, $T\subset B\subset G$ is a choice of Borel and maximal torus, and $W$ is the Weyl group, then
1) $H^{*}(G/B,K)=K[T^{\vee}]/(K[T^\vee]^W_{+})$
and
2) $K[T^\vee]^...

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### Characters of orthogonal groups as symmetric functions

This question was asked on MSE some time ago, here, but got no attention.
The Schur functions are characters of irreps of the unitary group, $s_\lambda(U)=Tr(R_\lambda(U))$. They are symmetric ...

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### Shimura varieties of Hodge type

I am trying to understand the theory of integral model of Shimura variety of Hodge type, like for example in Kisin's paper "Integral models for Shimura varieties of abelian type".
I understand that ...

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### Deformation of the Hochschild-Kostant-Rosenberg isomorphism for universal enveloping algebra

Let $\mathfrak{g}$ be a Lie algebra over a char. $0$ field and let $\iota: U\mathfrak{g}\rightarrow S\mathfrak{g}$ be the Poincaré-Birkhoff-Witt (PBW) isomorphism, inverse to that natural map from the ...

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### Hyperbolic Dehn surgeries and SU(2)-representations

Let $S^3-K$ be the complement of the figure eight knot complement. Thurston, in his Lecture Notes, constructed a hyperbolic structure, which comes from a discrete, faithful representation $\pi_1(S^3-K)...

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### $\text{Determinant}=(\sum \text{Determinant})^2$

Denote by $\delta_{n-1}=(n-1,n-2,\dots,1,0,0,\dots)$ the staircase partition and the embedded partition
$\lambda=(\lambda_1,\lambda_2,\dots)\subset\delta_{n-1}$.
QUESTION 1. Is this true?
$$\det\...

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### $q$-plane partitions & specialization & interlinks

MacMahon's enumeration of all plane partions (PP) inside an $n$-cube generalizes to
$${\tt PP_n}(q)=\prod_{i,j,k=1}^n\frac{1-q^{i+j+k-1}}{1-q^{i+j+k-2}}.$$
A $q$-analogue of symmetric plane partitions ...

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### Even Counterexample to Statement About the Non Existence of Certain Groups with Two Irreducible Monomial Character Degrees

Let $\textrm{cd}(G)=\lbrace \chi(1)\,|\, \chi\in\textrm{Irr}(G)\rbrace$ denote the set of character degrees of a finite group $G$. Similarly, denote by $\textrm{mcd}(G)$ the set of monomial character ...

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### Branching to Levi subgroups in SAGE and the circle action

In the SAGE computer package, there useful exist tools for branching representations of a simple Lie group to a Levi subgroup:
http://doc.sagemath.org/html/en/reference/combinat/sage/combinat/...

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### GAP versus SageMath for branching to Lie subgroups

Which computer package is better, GAP or SageMath, for
decomposing an irreducible representation of a (simple) Lie group
$G$ into representations of a Lie subgroup. I am most interested when
...

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### Classification of algebras of finite global dimension via determinants of certain 0-1-matrices

I restrict to the elementary problem that is equivalent to give a classification when Morita-Nakayama algebras have finite global dimension (see the end of this post for some background).
A Morita-...

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### Identity for classes of plane partitions

There are several classes of plane partitions in the literature.
Among these, let's look at the enumeration of three of them: the symmetric (SPP), totally symmetric (TSPP) and totally symmetric and ...

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### Derived invariance of the Cartan determinant

The Cartan matrix $C$ of a finite quiver algebra $A$ with points $e_i$ is defined as the matrix having entries $c_{i,j}=\dim(e_i A e_j)$. The Cartan determinant is defined as the determinant of the ...

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### The vanishing of sum of coefficients: symmetric polynomials

Denote $\pmb{X}_n=(x_1,x_2,\dots,x_n)$. Consider the symmetric polynomial
$$f_n(\pmb X_n)=\prod_{1\leq i<j\leq n}(x_i+x_j).$$
Expand these in terms of elementary symmetric polynomials, say
$$f_n(\...

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### 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 ...

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### Is it possible to reconstruct a finitely generated group from its category of representations?

Suppose $G$ is a finitely generated group, and suppose $Rep_k(G)$ is its category of representations over some field (or maybe even a ring) $k$, endowed with whatever extra structure is needed --- ...

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### Looking for access to original paper for Category O

It is well-known that the BGG category $\mathcal{O}$ was introduced in the early 1970s by Joseph Bernstein, Israel Gelfand and Sergei Gelfand. I google for a while but I cannot find out the original ...

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### Which dimensions exist for irreducible quaternionic-type real representations of finite groups?

I'm writing a software package to decompose group representations, and am struggling to find good examples of quaternionic-type representations of dimension > 4.
Reading MathOverflow, I found that ...

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### Littlewood-Richardson coefficients for zonal polynomials

The Littlewood-Richardson coefficients $c^\lambda_{\mu\nu}$ appear in the expansion of a product of Schur functions into Schur functions, $s_{\mu}(x)s_\nu(x)=\sum_\lambda c^\lambda_{\mu\nu}s_\lambda(x)...

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### Gorenstein symmetric conjecture for arbitrary rings

The Gorenstein symmetric conjecture states that for Artin algebras $A$ one has the the regular module has finite injective dimension as a right module if and only if it has finite injective dimension ...

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### Infinite-dimensional representation theory of $K[x]$

Let $K$ be an algebraically closed field. The finite-dimensional representation theory of the polynomial algebra $K[x]$ is tame and completely understood, which I shall first summarise. It's ...

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### Construction of non-split extension of simple modules of Lie algebras using linear differential operators

Consider the natural action of $W_1=k\left<x,\frac{d}{dx}\right>$ on $X=\mathbb C[x]$, then $\frac{d}{dx}, x\frac{d}{dx},x^2\frac{d}{dx}$ is essentailly a $\mathfrak{sl_2}$-tuple ($[x\frac{d}{dx}...

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### Does $SU(N)$ have pseudo-real representation?

For $N\ge 2$, does $SU(N)$ have a non-real pseudo-real irreducible representation? (The adjoint representation of $SU(N)$ is real).
A (complex, finite-dimensional) representation $R:SU(N)\to GL_n(\...

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### Frobenius formula

I know two formulas by the name of Frobenius.
The first one computes the number
$$\mathcal{N}(G;C_1,\dotsc,C_k):=|\{(c_1,\dotsc,c_k)\in C_1 \times \cdots \times C_k\:|\:c_1\cdots c_k=1\}|,$$
where $...

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### Counting points on connected reductive monoids over a finite field

Let $M$ be a connected reductive monoid over a finite field $\mathbb F_q$, is there an explicit formula of $\#M(\mathbb F_q)$?
There is something similar to Bruhat decomposition with the Weyl group ...

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### Which groups can be reconstructed from a single invariant subspace?

Let $G\subseteq\mathrm{Perm}(\Bbb R^n)$ be a matrix group consisting of permutation matrices acting on $\Bbb R^n$. Let $U\subseteq\Bbb R^n$ be an irreducible invariant subspace w.r.t. $G$. Now, define ...

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### Size of a multi-segment of a representation of $GL_n(F)$

Let $F$ be a p-adic field and $GL_n(F)$ the general linear group over $F$. The irreducible complex finite length smooth representations are parametrized by multi-segements in the paper. A multi-...

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### representations with centralizer stable under conjugate transpose

Let $\rho:G\to GL_n(\mathbb{C})$ be a finite-dimensional representation of a finite group $G$ over $\mathbb{C}$, and $C_\rho\subset M_n(\mathbb{C})$ its centralizer, i.e. $m\in C$ iff $m$ commutes ...

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### Algorithms for the explicit matrix isomorphism problem over $\mathbb{C}$

Suppose that $A$ is a $d^2$ dimensional algebra over $\mathbb{C}$ and we know the multiplication tensor $c_{ij}^k$ and the unit $u^k$ in some basis. If $A$ is semi-simple and has a single simple ...

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### Growth of the number of fixed points of a $p$-adic group under natural filtrations

Let $G$ be a $p$-adic reductive group, so as a locally profinite group it's the group of $\mathbb Q_p$ points of a connective reductive group over $\mathbb Q_p$, $K$ be a parabolic subgroup of $G$, ...

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### Closures of orbits in the space of representations of a quiver

Let $Q$ be a quiver, and let $d=(d_i)$ be a dimension vector. We can consider Rep($Q,d$), the affine space consisting of representations of $Q$ with dimension vector $d$. The general linear $GL(d)= \...

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### Periodic modules in Frobenius algebras

Let $A$ be a finite dimensional Frobenius algebra and assume there exists an indecomposable periodic module $M$, that is $\Omega^n(M) \cong M$ for some $n$.
Question: Does this imply that there is ...

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### The degree prime-power graph of $Suz$

Let $G$ be a finite group. Let $\mathrm{cd}(G)$ be the set of irreducible complex character degrees of $G$ and $\rho(G)$ the set of primes dividing degrees in $\mathrm{cd}(G)$. The authors define a ...

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### Translation of Soergel's 1990 paper on category O

Is there any English translation for the folowing paper of Soergel?
Kategorie $\mathcal{O}$, perverse Garben, und Moduln über den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990), 421-445,...

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### How to construct groups and large dimension representations? How about faithful ones?

Below I am referring to complex representations.
We know that if $G$ is a finite group with $m=(G:Z(G))$, then every irreducible representation has size at most $\sqrt{m}$. One cannot hope for this ...

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### Finding all $d$-dimensional indecomposable representations

Given a connected quiver algebra $A$ over a finite field $K$.
Question : Is there an effective/quick method to obtain all $d$-dimensional indecomposable representations for a fixed $d$ with a ...

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### SL(2, C)-representation of a knot

When studying knot theory I often encounter $SL(2, \mathbb{C})$-representation of knots (of the knot group) or the $SL(2, \mathbb{C})$ character variety of a knot group. But I just don't seem to ...

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### Example of a finite group $G$ with low dimensional cohomology not generated by Stiefel-Whitney classes of flat vector bundles over $BG$

In Stiefel-Whitney classes of real representations of finite groups, J. Algebra 126 (1989), no. 2, 327–347, Gunawardena, Kahn and Thomas dealt with the question, whether the cohomology ring $H^...

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### Constructive proof of Swan theorem

Let $M$ be an $S_n$-lattice (so it is free as an abelian group), and assume that $M$ is projective (i.e. direct summand of some $\mathbb Z[S_n]^m$). A theorem of Swan implies that $M$ is stably ...

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### Failure of Schur's lemma for topological group representations

Is there an example of $G$, $\rho$ as below?
$G$ is a locally compact group.
$\rho$ is an irreducible continuous representation of $G$ on a complex Hilbert space $V$. This means that we have a ...

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### Calculate the character degrees of a finite group $G$

Let $G$ be a finite group and $K$ be a group of order $8$. Suppose that $G/K\cong M_{12}$ where $M_{12}$ is one of the Mathieu groups.
QUESTION:
How to calculate the all complex character degrees ...

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### Reference Request: Structure constants for G2

Let $G$ be a split semisimple real Lie group in characteristic zero, and let $B=TU$ be a Borel subgroup with unipotent radical $U$ and Levi $T$. Fix an ordering on the roots $\Phi^+$ of $T$ in $U$, ...