Questions tagged [rt.representation-theory]

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

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Homology of an interesting Lie algebra

Let $U$ and $V$ be finite dimensional complex vector spaces (or perhaps graded vector spaces). Let $E(U)$ be the "square zero extension" $\mathbb C \oplus U$, made into a commutative ring in such a ...
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Is there a reason nice coset representatives exist for Leech or E_8 lattice modulo 2?

Let $\Lambda$ be the Leech lattice. There is a nice set of coset representatives for $\Lambda/2 \Lambda$ given by short vectors [Conway and Sloane, Ch. 10, Theorem 28 or Ch. 23, Theorem 3]. The proof ...
Tathagata Basak's user avatar
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Littlewood-Richardson sequences and Littlewood-Richardson coefficients

I'm looking for a proof or a reference for the following statement, I give the definitions below: There exists a Littlewood-Richardson sequence of type $(\alpha, \beta, \lambda)$ if and only if $...
Maffred's user avatar
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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
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When do almost all these invariants of tensors vanish?

Let $A,B,C,D$ be $n$-dimensional vector spaces over a field $k$. There is a natural homomorphism from the $mn^m$th tensor power $A^{\otimes (m n^m)} $ of $A$ to $k$ given by the determinant map $A^{\...
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pattern-avoiding permutations vs multi-core partitions

Let $\mathfrak{S}_n$ be the permutation group on $[n]$. Given the pattern $\sigma=k(k-1)\cdots321$, let $I_n(\sigma)$ be the number of involutions in $\mathfrak{S}_n$ that avoid the pattern $\sigma$. ...
T. Amdeberhan's user avatar
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How do sutured TQFT fit into the larger TQFT picture?

In https://arxiv.org/abs/0807.2431, Honda--Kazez--Matic introduce a definition of (1+1 dimensional) sutured TQFT; see also e.g. Mathews http://arxiv.org/abs/1006.5433 and Fink https://arxiv.org/abs/...
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Software for explicit computations in representations of classical Lie algebras

I'm pretty sure many a mathematician has longed for such a tool but I wasn't able to find such a question here, so here we go. Is there, by any chance, an existing package or program that allows one ...
Igor Makhlin's user avatar
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How to show the compatibility between Duflo isomorphism and Harish-Chandra isomorphism for semi-simple Lie algebras?

I was told that the Duflo isomorphism is compatible with the Harish-Chandra isomorphism when the Lie algebra $\mathfrak{g}$ is semi-simple. However I cannot see why this is true. All I can show is ...
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What's the analogue of a Young symmetrizer in the Brauer algebra?

According to Schur--Weyl duality, the centralizer of $\mathrm{GL}(V)$ acting diagonally on $V^{\otimes N}$ is the group algebra of the symmetric group $\mathbb S_N$. An equivalent formulation is the ...
Dan Petersen's user avatar
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Intersection of Springer fibre and Schubert cell

Let us consider intersections of Springer fibres and Schubert cells in type A. Let $ Y : \mathbb C^n \rightarrow \mathbb C^n $ be a nilpotent operator. Let $$ F_Y = \{ V_0 = 0 \subset V_1 \subset \...
Joel Kamnitzer's user avatar
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The space-time dimension of the N-superstring theory?

Let $\mathfrak{W}$ be the Lie algebra generated by $d_{n} = ie^{in\theta}\frac{d}{d\theta}$ and $\mathfrak{Vir} = \mathfrak{W} \oplus C \mathbb{C}$ its central extension: $$ [L_m,L_n]=(m-n)L_{m+n}+\...
Sebastien Palcoux's user avatar
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Degree of a cone over the set of rank $r$ $n\times n$ matrices

Let $X_r\subset Mat_{n\times n}(C)$ denote the variety of rank at most $r$ matrices, set $k=n-r$ and assume $n\geq k^2-1$. Consider the cone over $X_r$ with vertex spanned by the first $k^2-1$ entries ...
JM Landsberg's user avatar
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Quantum Drinfeld-Sokolov reduction for a module

There is a well-established procedure for quantizing the Drinfeld-Sokolov reduction for an affine Lie algebra. In particular, this paper of de Boer and Tjin describes an algorithm to produce the ...
Christopher Beem's user avatar
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Why is this vector space related to modular forms?

In the course of doing some calculations on a project I am working on, I came across the following presentation of a vector space. It is generated by homogenous polynomials of even degree $n$ over a ...
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Twisted Springer fibers

In the study of certain moduli spaces of $p$-divisible groups I came across the following twisted version of a Springer fiber, and I was wondering whether some expert on algebraic groups/algebraic ...
Torsten Wedhorn's user avatar
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Status of Borho and Brylinski's irreducibility conjectures?

In Differential Operators on Homogeneous Spaces, III, Section 6.7, Borho and Brylinski make a long series of equivalent conjectures which they show are all equivalent. Perhaps the easiest statement ...
Ben Webster's user avatar
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Role of nontrivial component groups in Springer Correspondence?

Set-up for classical Springer Correspondence: $G$ = reductive group (usually assumed to be semisimple of adjoint type) over $\mathbb{C}$, with Borel subgroup and maximal torus $B \supset T$, Weyl ...
Jim Humphreys's user avatar
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An identity for characters of the symmetric group

I am looking for a reference for the identity $$\chi_\lambda(C)=\frac{\dim(V_\lambda)}{|C|}\sum_{p\in P_\lambda,\,q\in Q_\lambda,\,pq\in C}\operatorname{sgn}(q)$$ for the irreducible characters of the ...
Hjalmar Rosengren's user avatar
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Is this set, defined in terms of an irreducible representation, closed under inverses?

$\DeclareMathOperator\GL{GL}$Let $ H $ be an irreducible finite subgroup of $ \GL(n,\mathbb{C}) $. Define $ N^r(H) $ inductively by $$ N^{r+1}(H)=\{ g \in \GL(n,\mathbb{C}): g H g^{-1} \subset N^r(H) \...
Ian Gershon Teixeira's user avatar
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Singularity category of a hypersurface associated to $M_{11}$

For reasons to do with classifying spaces of finite groups, I have the following algebra. Let $k$ be a field of characteristic two, and let $R = k[x,y,z]/(x^2 y + z^2)$, as a graded $k$-algebra with $|...
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Alternate proof in Fulton–Harris of representation theoretic version of Littlewood–Richardson rule

$\DeclareMathOperator\Ind{Ind}$Let $d = d_1 + d_2$ with $d_1$, $d_2$ positive integers. Let $\lambda$ be a partition of $d_1$ and $\mu$ a partition of $d_2$, so that the Young symmetrizer construction ...
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Interpretation of $p$-adic 'smoothness'

real case: In the very first course of Calculus, one learns that a real function $f \colon \mathbb{R} \to \mathbb{R}$ is called smooth, if it is differentiable as many times as one pleases. So the ...
Maty Mangoo's user avatar
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Semigroups of matrices closed under conjugate transposition

An involution semigroup or $\star$-semigroup is a unary semigroup $\langle S,{\cdot}\,,{}^\star\rangle$ that satisfies the equations $$ (x^\star)^\star = x \quad \text{and} \quad (xy)^\star = y^\star ...
E W H Lee's user avatar
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Is the Lawrence–Krammer representation faithful, reduced modulo p?

It is well-known that the braid group $B_n$ is linear for every $n$ by the Lawrence–Krammer (or LKB) representation. It embeds $B_n$ faithfully into $\mathrm{GL}\left(\frac{n(n-1)}{2},\mathbb{Z}[q^{\...
Adel M's user avatar
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Finite-dimensional representations of quantum $SU(2)$

The most famous of all the quantum groups is $SU_q(2)$ - the Quantum special unitary group. The irreducible comodules of this quantum group are very well understood - they are labelled by integers (or ...
Jake Wetlock's user avatar
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Why are these Littlewood-Richardson coefficients congruent to 1 mod 8?

Let $n\in{\mathbb N}$ and write $n=q_1+q_2+\dots+q_t$, where $q_1>q_2>\dots>q_t$ are powers of $2$. Let $\lambda_n$ be the partition with Frobenius symbol $(q_1-1,q_2-1,\dots,q_t-1;q_t,q_{t-1}...
John Murray's user avatar
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Scalar products on symmetric functions behaving like the Macdonald scalar product

The Macdonald symmetric functions (or Macdonald polynomials) $P_\lambda(x)$ are orthogonal with respect to the Macdonald scalar product $$ \langle p_\lambda,p_\mu\rangle = \delta_{\lambda\mu}z_\...
Richard Stanley's user avatar
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Sum of Littlewood-Richardson coefficients

(Asked first in MSE) Let $c_{\lambda\mu}^\nu$ be the usual Littlewood-Richardson coefficients, which appear for example in the expansion of a product of Schur polynomials into Schur polynomials, $$ s_\...
thedude's user avatar
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A Lie-theoretic question regarding $B\ltimes \mathfrak{g}/\mathfrak{b}$

I am stuck on a seeming elementary Lie-theoretic question arising from a study of components of affine Springer fibers. Will be very grateful if somebody would like to share some insight, or ...
Cheng-Chiang Tsai's user avatar
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Image of multiplication map in tensor powers of finite-dimensional ring

Let $R$ be a (commutative, unital) ring of dimension $n$ over a field $k$. Assume the characteristic of $k$ is greater than $n$. Then $R^{\otimes n}$ has a natural ring structure, together with an $...
Will Sawin's user avatar
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Equivariant coherent sheaf category for unipotent group actions

Suppose $U$ is a complex algebraic unipotent group. Let $X$ be a projective variety with a $U$-action. For simplicity, we may assume that there are only finite many $U$ orbits on $X$. The primary ...
Amanda Taylor's user avatar
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Formal degree of discrete series representations

Let $G$ be a locally compact unimodular group. A continuous irreducible unitary representation $\pi$ of $G$ is said to be a discrete series if its matrix coefficients (defined by $\xi^\pi_{v,w} : g \...
Desiderius Severus's user avatar
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Representation theory of Chevalley groups as a categorical trace

Dennis Gaitsgory's 2016 preprint, From Geometric to Function-Theoretic Langlands (or How to Invent Shtukas) includes in the third section a very compressed but suggestive discussion of the ...
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Relation between the Laplace eigenvalue of Maass forms and the Virasoro algebra

I attended today a talk of Dorian Goldfeld. In the talk, he mentioned that for a Maass cusp form $\phi$ of $\mathrm{SL}(n, \mathbb{Z})$, with Langlands parameter $\alpha = \left( \alpha_1, \dots, \...
darkl's user avatar
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Name for an involution associated to a Coxeter element

Let $(W,S)$ be a finite Coxeter system, and $c \in W$ a Coxeter element. There is an involution $g\in W$ for which the involutive map $w \mapsto gw^{-1}g$ fixes $c$. Is there a standard name for this ...
Sam Hopkins's user avatar
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When does a semisimple $\mathbb{C}$-algebra come from a group?

Let $\mathcal{A}$ be a semisimple $\mathbb{C}$-algebra. By the Artin-Wedderburn theorem, it is isomorphic to a direct product of matrix algebras: $$ \mathcal{A} = \prod_{i=1}^m M_{n_i}(\mathbb{C})$$ ...
pitariver's user avatar
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references on categorification of knot invariants

I am extremely sorry if this is not the right place for this kind of question. I have studied some knot theory, quantum invariants and would like to study more about categorification of knot ...
Jernej Grlj's user avatar
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Dyck paths of Dynkin type

(The conjecture is a homological algebra question, but question 2 is a pure combinatorics question given that the conjecture is true) A Dyck path of length $n$ is a list of positive integers $[c_1,c_2,...
Mare's user avatar
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Generalized Young symmetrizers, why not?

For $n$ a positive integer, let $[n]=\{1,2,\ldots,n\}$. Consider two set partitions $\mathcal{A}=\{A_1,\ldots,A_p\}$ and $\mathcal{B}=\{B_1,\ldots,B_q\}$ of the set $[n]$. We will denote by $G(\...
Abdelmalek Abdesselam's user avatar
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Computing the number of elementary abelian p-subgroups of rank 2 in $GL_{n}(\mathbb{F}_{p})$

Let $p$ be a prime number, and let $\mathbb{F}_{p}$ be a finite field of order $p$. Let $GL_{n}(\mathbb{F}_{p})$ denote the general linear group and $U_{n}$ denote the unitriangular group of $n\times ...
Nourddine Snanou's user avatar
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Two algebraic guises of Alternating Sign Matrices: any connection?

Alternating Sign Matrices (ASMs) have a famous history: they were discovered by Mills, Robbins, and Rumsey, who conjectured a product formula for their enumeration; this product formula was first ...
Sam Hopkins's user avatar
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Non-algebraic representations of $\text{SL}_n(\mathbb{R})$

My question is easily stated: are all continuous finite-dimensional real representations of $\text{SL}_n(\mathbb{R})$ algebraic representations? This is false if you drop the word "continuous" (e.g. ...
Tina's user avatar
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Non-zero group determinant for symmetric group

Let $G$ be a finite group. Given complex numbers $x=\{x_g: g\in G\}$, one can define a $|G|\times |G|$ matrix $X$, with entries $X_{g,h} = x_{gh^{-1}}$. Let's consider $G$ being the symmetric group $...
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Are the braid groups good in the sense of Toën?

In this question it is asked whether the mapping class groups are 'good' in relation to pro-finite completion. Helpfully, one of the answers gives a link to a proof that the braid groups are good ...
Patrick Elliott's user avatar
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What are the character tables of the finite unitary groups?

I need to know the (complex) character table of the finite unitary group $U_n(q)$. Lusztig and Srinivasan (1977) provide an abstract description, but parsing it requires a stronger background in ...
Dustin G. Mixon's user avatar
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Significance of half sum of non-simple positive roots

In representation theory, there are plenty of places that a $\rho$-shift makes an appearance, where $\rho$ is the half sum of positive roots. See, for instance, this post for some discussions of the ...
Dr. Evil's user avatar
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Which representations of the Lie algebra of a Lie group come from representations of the group itself?

I think this is very classic mathematics, but I can't find a complete answer in the literature. Let $G$ be a Lie group, $\mathfrak{g}$ the Lie algebra of $\mathfrak{g}$. Suppose $\rho : \mathfrak{g} \...
Rick Sternbach's user avatar
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$n$-fold tensor products of $D(A)$ for finite dimensional algebras

Let $A$ be a finite dimensional quiver algebra over a field $K$ and let $D(-):=Hom_K(-,K)$ denote the natural duality (assume algebras are connected). Define $\psi_A:=sup \{ n \geq 1 | D(A)^{\otimes ...
Mare's user avatar
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On constructible Hall algebra and instantons

I heard in a talk by Yan Soibelman that by starting with a quiver $Q$ with a set of vertices $I$ we can either symmetrize or anti-symmetrize its Euler-Ringel form $\chi_Q$. He claims that anti-...
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