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

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A problem with pointwise stabilizer subgroups of fixed-point subspaces II

Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$. Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$. Let ...
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1answer
300 views

Constant group scheme and torsors

Let $X$ be a scheme and $G$ a (commutative) constant group scheme. Consider a $G$-torsor $Y$ for $X$, by which I mean that there is a canonical isomorphism: $$g_Y \colon Y \times_X Y \cong Y ...
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1answer
61 views

Reference that contains examples of absolutely indecomposable representations of quivers over a finite field

Is there a reference that lists/discusses examples of absolutely indecomposable representations of quivers over a finite field (absolutely indecomposable = does not decompose into a direct sum over ...
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58 views

Differences between primitive central idempotents and primitive orthogonal idempotents

If we have a complete set of primitive orthogonal idempotents of an algebra $A$, then we can obtain simple modules, indecomposable projective modules, indecomposable injective modules of $A$. If we ...
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1answer
140 views

Represent matrix immanants using Schur functions

For each irreducible character $\chi^\lambda$ of the symmetric group $S_n$, the immanant of an $n\times n$ square matrix $A$ is defined as \begin{equation*} d_\lambda(A) := \sum_{\sigma \in S_n} ...
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A problem with pointwise stabilizer subgroups of fixed-point subspaces I

Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$. Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$. Let ...
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1answer
115 views

Beilinson-Bernstein localization: $\mathfrak{g}$ action on $G$-equivariant sheaf

I have a few elementary questions related to Beilinson-Bernstein localization. Let $G$ be a semisimple algebraic group over $\mathbb{C}$ with Lie algebra $\mathfrak{g}$. Consider the setup of ...
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Is an $\mathfrak{sl}_2$-triple determined up to Lie algebra automorphism by the adjoint representation?

Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra, and let $\phi_1:\mathfrak{sl}_2(\mathbb{C})\rightarrow\mathfrak{g}$ and ...
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774 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 ...
3
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0answers
323 views

A dual version of a theorem of Øystein Ore in group theory

Let $(H \subset G)$ be an inclusion of finite groups. This post is a dual version for the Generalization of a theorem of Øystein Ore in which it's proved: Theorem: $\mathcal{L}(H\subset G)$ ...
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38 views

Intertwining Operators Associated to Simple Reflections

Let $G$ be a quasi-split reductive group, over a local field, with a Borel subgroup $B=T\cdot N$ and the associated Weyl group $W$. Given a family of induced representations $\pi_s = Ind_B^G \chi\cdot ...
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Generators of invariant polynomials of semisimple Lie algebra

Suppose $\mathfrak{g}$ is a complex semi-simple Lie algebra. By a theorem of Chevalley, we know that $S(\mathfrak{g})^\mathfrak{g}$, i.e. the $\mathfrak{g}$ invariant polynomials, is generated by $l$ ...
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1answer
91 views

Prescribed spherical representations, symplectic group $Sp(n)$

An irreducible representation $(\pi,V_\pi)$ of a compact group $G$ is called spherical with respect to the pair $(G,K)$, $K$ is closed subgroup of $G$, if $V_\pi$ has a non-zero vector invariant by ...
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0answers
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References for the bicategory of ring-bimodule pairs

One of the standard examples of a bicategory is the bicategory of rings (with bimodules as 1-morphisms), which is sometimes denoted $\operatorname{Bim}$ and in other sources $\operatorname{Ring}$ (or ...
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1answer
223 views

Is there a nonabelian finite simple group with Grothendieck ring of multiplicity one?

Let $G$ be a finite group. It admits finitely many irreducible complex representations $H_1, \dots, H_r$ which generate, for $\oplus$ and $\otimes$, the Grothendieck ring $\mathcal{G}(G)$ of $G$ (also ...
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607 views

Are there any natural differential operators besides $d$?

Let $\lambda = (\lambda_1, \ldots, \lambda_r)$ and $\mu = (\mu_1, \ldots, \mu_r)$ be partitions such that $\mu_j = \lambda_j +1$ for one index $j$ and $\mu_i = \lambda_i$ for all other $i$. Then there ...
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Why Jacobson, but not the left (right) maximals individually?

I firstly asked the following question on MathStackExchange, but I did not receive any responses, but a short comment. So, I decided to post it here, hoping to receive answers from experts. ...
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Fun applications of representations of finite groups

Are there some fun applications of the theory of representations of finite groups? I would like to have some examples that could be explained to a student who knows what is a finite group but does not ...
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485 views

Recommendation textbooks on D-module

I am going to take part in a seminar on D-modules and applications, the textbooks that will be used are : D-modules, Perverse Sheaves, and Representation Theory, A Primer of Algebraic D-Modules ...
7
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1answer
300 views

On a theorem of Kazhdan

Let $G=GL_n(F)$, where $F$ is a p-adic local field, $U$ be the upper triangular maximal unipotent group, and $\theta$ a character of $U$. Then a Theorem of Kazhdan says that for any irreducible smooth ...
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3answers
164 views

how to find explicitly given component in a regular representation

Given a finite group $G$ and its irreducible representation $\pi$ I want to find explicit elements of the group algebra $\mathbb{C}[G]$ lying in components of the left regular representation ...
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$\Gamma$ cohomology of principal series

Let $G$ be a noncompact connected real semisimple Lie group with finited center. Let $\Gamma$ be a cocompact discrete subgroup of $G$, and let $P$ be a parabolique subgroup with Langlands ...
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1answer
209 views

Finite field analogue of representations in same packet have equal central character

In Kevin Buzzard's recent question, a warm up question was: if two automorphic representations are nearly equivalent, then are the central characters of their local components equal? Working my way ...
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1answer
232 views

On a drawing in Dixmier's Enveloping Algebras

This image comes from Dixmier's book, 'Enveloping Algebras' ('Algèbres enveloppantes'). Dixmier writes that The curves shown on p. XIV have their origin in the study of U(sl(3)). They are ...
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1answer
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Matrix Elements of Real Representations

I asked this question over at Math.StackExchange and despite having had a bounty on it I did not receive an answer. Suppose that $G$ is a finite group and we have a unitary irreducible representation ...
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1answer
957 views

When does a perverse sheaf occur in the decomposition theorem?

Suppose I am in the setting of the decomposition theorem, i.e., we have the decomposition of the direct image $f_*\mathbb Q_\ell$, where $f:X\to Y$ is proper. Then the direct image decomposes into a ...
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1answer
476 views

Which L-functions are not “Langlands-Shahidi L-functions”?

The Langlands-Shahidi method, among other things, obtains certain L-functions from the constant term of Eisenstein series attached to so-called $(G,M)$ pairs, where $G$ is a reductive group, $M$ a ...
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41 views

Framed braids and local systems

Let me start by admitting that my question is going to be somewhat vague. But hopefully it is one of these vague questions that can be immediately answered by an expert in the appropriate area. ...
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0answers
123 views

representation of $SO(p)\times SO(q)$ with $p,q$ odd

Assume $p,q$ odd. We denote by $\sigma_p$ the standard representation of $SO(p)$, that is the representation of $SO(p)$ acting on $\mathbf{R}^p$ as matrix. So is $\sigma_q$. Take $K=SO(p)\times ...
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What is the name for the subring of the Grothendieck ring of a bialgebra spanned by one-dimensional representations?

Let $B$ be a finite dimensional bialgebra over a field $\Bbbk$. Let $\mathcal G_0(B)$ be the ring whose underlying additive group is generated by isomorphism classes $[V]$ of finite dimensional ...
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1answer
171 views

Applying Lemma 2.12 of Deligne's/ Milne's “Tannakian Categories” on an irreducible representation

since this is my first question here, I'm not very certain whether I state it properly. I'm thankful for any helpful remarks. Currently I'm trying to understand the above-mentioned article which can ...
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1answer
377 views

Condition on a Hopf operad for tensor product in the base category to be a (categorical) coproduct for algebras

A Hopf operad will be an operad endowed with a coproduct $P(n) \longrightarrow P(n) \otimes P(n)$ which is compatible in the obvious sens with operad laws (no more structure is assumed a priori. ...
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85 views

What interesting things do automorphism groups of trees act on?

Let $T$ be a rooted tree. We can build a poset $P(T)$ whose elements are the vertices of $T$ and whose covering relations are the edges of $T$. Let $A$ be the automorphism group of $P(T)$. The group ...
3
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1answer
124 views

About the second largest adjacency eigenvalue of Abelian Cayley graphs

[Assume all groups are finite] One knows the general statement that the sum of the values of the character function on the generating set is an eigenvalue of a Cayley graph. But the above doesn't ...
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3answers
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Nice algebraic approximations of classifying spaces

Let $G=GL_k(\mathbb C)$ be the complex linear group. Then the infinite Grassmannian is a model for the classifying space $BG$. We can write the infinte Grassmannian as a colimit of the finite ...
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How can you see the minimal relations on a quiver from its bimodule resolution?

Suppose that you are given an algebra $KQ/I$, coming from a quiver Q, of finite global dimension. Suppose also that you know its minimal bimodule resolution over its enveloping algebra. Can you get a ...
2
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1answer
132 views

Invariant subspaces of an $F_2$-representation of the affine linear group of dimension 1

Let $p$ be an odd prime (large if it matters) and let $G= Aff(\mathbb{F}_{p^2}) \cong \mathbb{F}_{p^2} \rtimes \mathbb{F}_{p^2}^*$ be the affine linear group acting on $\mathbb{F}_{p^2}$ by $x\mapsto ...
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104 views

Decomposition of symmetric powers of reduced regular representation modulo $p$

Let $\bar{\rho}$ denote the reduced regular representation of $\mathbb{Z}/p$ over a field of characteristic $p$. The representation $\mathrm{Sym}^k \bar{\rho}$ decomposes (for each $k$) as a sum of ...
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3answers
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Kaplansky's 6th conjecture: dim(Irrep) | dim(algebra) - for semi-simple Hopf algebras

Let $H$ be a semisimple Hopf algebra. One of the Kaplansky's conjectures states that the dimension of any irreducible $H$-module divides the dimension of $H$. In which cases the conjecture is known ...
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1answer
694 views

Number of standard Young tableaux with fixed corner entry

For a partition $\lambda=(\lambda_1\geq \lambda_2\geq\ldots\geq \lambda_k)$ of $n$, let the set of standard Young tableau of shape $\lambda$ be denoted by $SYT(\lambda)$ with boxes at $(i,j)$ denoted ...
4
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1answer
119 views

Motivational ideas for the Gelfand-Graev character of a finite group of Lie type

I've been studying the Gelfand-Graev character's general construction for a finite group of Lie type. I wish to discuss its particularization in a seminar for the general linear group over a finite ...
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142 views

Is there a method to simultaneously block-diagonalize a set of group matrices?

Assume that you are explicitly given the representation matrices of a group. How does one go about finding that common basis which will find the irreducible components of all of them simultaneously? ...
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Is the Duflo polynomial conjecture open?

Let $G/K$ be a symmetric space. Let $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ be a Cartan decomposition, with the odd part $\mathfrak{p}$. It is well known that the algebra of invariant ...
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1answer
257 views

Derivation of Blattner's conjecture in the Beilinson-Bernstein picture

On the last page of Schmid's article "Discrete Series", he says "In the Beilinson-Bernstein picture, discrete series modules are attached to closed $K$-orbits in $X$... the $K_{\mathbb R}$-structure ...
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Computing equivariant K-theory using the amalgamted product

If I have a Lie group (or a Kac-Moody group) $G$ such that it's the amalgated product of it's proper parabolic subgroups $P_J$, i.e. $G = \text{colim} P_J$, then could I use this to compute the ...
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1answer
160 views

What is the cohomology of the tangent bundle of a flag variety?

Let $G$ be the general linear group $\operatorname{GL}(n,\mathbb{C})$ and $P$ a parabolic subgroup with Lie algebra $\mathfrak{p}$. Consider the vector bundles $$ \mathcal{P} = G\times_P ...
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3answers
371 views

First Explicit Irreducible Representations

Although the classification of simple Lie Algebras and their representations is fully understood, I wonder whether there is some book with exhaustive tables describing explicit irreducible ...
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107 views

Dimension of Birman-Murakami-Wenzl Algebra

I was reading the paper Braids, Link Polynomials and A New Algebra by J. S. Birman and H. Wenzl, and I was wondering is there a combinatorial way to compute the dimension of the algebras ...
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Irreducible decomposition of $\Lambda^i(\mathfrak{p})$

Let $G$ be a connected semisimple real Lie group with finite center with Lie algebra $\mathfrak{g}$. Take $K$ its maximal compact subgroup of $G$, and $\mathfrak{k}$ its Lie algebra. We denote by ...