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

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7
votes
1answer
130 views

Cominuscule property of nilpotent orbits

Let $G$ be a complex reductive Lie group, $G/P$ a flag manifold, and $\Phi: T^* G/P \to {\mathfrak g}^*$ the moment map. So $\Phi(T^* G/P)$ is the closure of a nilpotent orbit. Lots of classes of ...
2
votes
0answers
60 views

Set of Special Unitary Matrices that are dense in SU(4) and obey certain relations

I'm trying to find a finite set of 4x4 Unitary matricies $\{U_1,U_2,\ldots U_N\}$ such that the matrices are dense in SU(4), and obey the relations: $[U_i, U_j] = 0$ for $|i-j|>1$ ...
2
votes
2answers
150 views

odd length Chevalley relations (in rank two)

The unipotent radicals $\text{N}$ of the Borel subgroups of the complex algebraic groups of type $A_2$, $B_2$, and $G_2$ can each be abstractly presented using two one-parameter subgroups $x_1, x_2: ...
12
votes
1answer
83 views

An inequality for the minimal number of generators of a finite group

Let $G$ be a finite group, $n(G)$ the minimal number of generators and $m(G)$ the minimal number of irreducible complex representations generating (with $\otimes$ and $\oplus$) the left regular ...
2
votes
0answers
62 views

Lie Symmetries of the Bessel Differential Equation

The Bessel differential equation has an arbitrary looking form, but a lot is known about it. $$ x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - n^2)y = 0 $$ Is there a way to derive the Bessel ...
2
votes
0answers
69 views

extending local systems on a neighbourhood

Let $Y$ an affine finite type scheme over an algebraically closed field $k$. Let $S$ be a closed subscheme of $Y$ and $Y'$ the henselization of $Y$ along $S$. If we have a $\mathbb{Z}_{\ell}$ local ...
1
vote
1answer
125 views

l-adic local system. on hensel schemes

Let $k$ be a field, $\ell$ a prime different from the characteristic. If I take $S$ a closed subscheme of $Y$, which is a $k$-scheme of finite type, is it true that any $\mathbb{Z}_{\ell}$-local ...
6
votes
0answers
93 views

Is there a finite test for isomorphisms of rigid monoidal abelian categories, part II

Let $G$ be a semisimple algebraic group. (I'm already interested in the case $G=SL_2$.) Let $\mathcal C$ be a semisimple rigid monoidal abelian category endowed with an additive tensor functor $Rep_G ...
1
vote
1answer
222 views

Reference request for an introduction to deformation theory in algebraic geometry

I'd like some introductory references for deformation theory in algebraic geometry. I'm interested in survey articles too but I primarily want references which give all the definitions and go through ...
12
votes
1answer
163 views

Is there a finite test for isomorphisms of rigid monoidal abelian categories?

Let $G$ be a semisimple algebraic group. (I'm already interested in the case $G=SL_2$.) Let $\mathcal C$ be a semisimple rigid monoidal abelian category endowed with pair of exact tensor functors ...
22
votes
6answers
3k views

Nice proofs of the Poincaré–Birkhoff–Witt theorem

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra with an ordered basis $x_1 < x_2 < ... < x_n$. We define the universal enveloping algebra $U(\mathfrak{g})$ of $\mathfrak{g}$ to be the ...
13
votes
1answer
388 views

Most discriminants are almost squarefree

Write, for $f(x) = x^d + a_2 x^{d-2} + \cdots + a_d\in \mathbb{Z}[x]$, $H(f) := \max(|a_i|^{\frac{1}{i}})$. Does anyone know of a reference that would allow me to show that the proportion of $f$ with ...
7
votes
1answer
165 views

Invariants and stablizers for the $PGL(V)$ action on $End(V\otimes V)$

Let $K$ be a field of characteristic zero, and $V$ a finite dimensional vector space over $K$. Consider the action of the algebraic group $G:=PGL(V)$ on the vector space $W:=End_K(V^{\otimes 2})$ by ...
4
votes
1answer
312 views

Birch's conjecture from Representation Theory

Birch has a conjecture about which automorphic forms on $PGL(2)$ are the lifts from nonsplit $O(3)$. Temporarily ignore global issues, and focus on the local nonarchimedian picture. The automorphic ...
1
vote
0answers
96 views

Conjugacy scheme, fppf versus GIT

I would be glad to have some guidance in the following. Let $k$ be an algebraically closed field. Let $G$ be a connected reductive group over $k$. Denote by $\mathfrak{c}$ the Zariski spectrum of the ...
2
votes
1answer
163 views

Sampling from random totally unimodular matrices of a particular type?

Is there a way to parametrize totally unimodular $(3n+2)\times(2n+2)$ matrices of form $$\begin{bmatrix} \pm1 & \pm1 & 0 & 0 &\dots & 0 & 0 & 0 & 0\\ A_{2n} & ...
6
votes
0answers
67 views

Subquotients of Jantzen Filtration for Kac-Moody algebras

Let $\mathfrak{g}$ be a complex symmetrizable Kac-Moody algebra, with triangular decomposition $\mathfrak{n}^- \oplus \mathfrak{h} \oplus \mathfrak{n}^+$. Let $\lambda \in \mathfrak{h}^*$, and ...
6
votes
0answers
99 views

Local character expansion for discrete series representations of $GL_n(F)$

I'm interested about what, if anything, is known about the local character expansion of discrete-series representations of $GL_n(F)$, where $F$ is a $p$-adic field. First, some notation: let $G$ be a ...
5
votes
1answer
95 views

Does a $G$-structure on $M$ automatically descend to a contact $G$-structure on $\mathbb{P}T^*M$?

If $M$ is a real smooth manifold of dimension $n+1$, by $D\in\mathbb{P}T^*M$ I mean a tangent hyperplane at some point of $M$. I denote by $b$ the canonical projection of the $(2n+1)$-dimensional ...
4
votes
0answers
137 views

Representation theory and associated bundles

I am looking for a text or set of notes that discusses the relationship between the representation theory of Lie groups and associated vector bundles, preferably using modern categorical language. For ...
7
votes
1answer
697 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: $\mathcal{L}(H\subset G)$ ...
5
votes
2answers
170 views

Center of quantum affine algebras

Are there some references about the center of quantum affine algebras? I searched on google and only find the paper. In particular, what is the center of $U_q(\widehat{\mathfrak{sl}_2})$. Thank you ...
3
votes
1answer
172 views

Affine analog of the theory of sheets

In the study of adjoint orbits in a complex semi-simple lie algebra, there is a well known object known as a "sheet". These are the irreducible components of the union of orbits of the same dimension. ...
3
votes
1answer
259 views

Reference request - Jacquet module and asymptotic of matrix coefficients

Hello, I would like to know some nice references about the relation between asymptotics of matrix coefficients of representations of reductive groups over local fields, and the pairing between the ...
16
votes
5answers
521 views

Is there a formula for the Frobenius-Schur indicator of a rep of a Lie group?

Let $G$ be a simple algebraic group group over $\mathbb C$. Let $V$ be a self-dual representation of $G$. Let $\lambda$ be the highest weight of $V$. Write $\lambda$ as a sum of fundamental weights: ...
3
votes
0answers
156 views

Finding density of Haar measure relative to the Liouville volume measure on $\frac{G^{\mathbb{C}}}{P}$

Let $G$ be a connected compact Lie group and $G^{\mathbb{C}}$ be the complexification of the Lie group $G$ then we know that by polar decomposition we can write $G^{\mathbb{C}}\cong G\times ...
11
votes
2answers
387 views

actions of the hyperoctahedral group

I am looking for actions (i.e., permutation representations) of the hyperoctahedral group $H_n$ (also known as the group of signed permutations) studied in the literature, i.e., homomorphisms from ...
3
votes
1answer
184 views

Is the Veronese variety “enough” to describe all the $SL(V)$-orbits in $\mathbb{P}(\textrm{Sym}^dV)$?

I apologise in advance if the question will look ridicolous to experienced eyes: in this case a good reference will be enough to clarify my doubts. Let $V$ be a complex vector space of dimension $n$, ...
3
votes
1answer
143 views

Existence of $SO(n)$-isotropic inner products which are not $O(n)$-isotropic

$\newcommand{\al}{\alpha}$ Let $M_n$ be the space of $n \times n$ real matrices. Question: For which $n$, is there an inner product on $M_n$ which satisfies: $$(*) \, \, \langle Q^TXQ,Q^TYQ ...
9
votes
3answers
232 views

Construct discrete series of SL(2,R) as kernel of twisted Dirac operators

I’m studying the paper of (Baum-Connes-Higson, ex 4.25), and I would like to give an explicit computation for the Connes-Kasparov conjecture for SL(2,R). The idea is that each non-trivial ...
16
votes
2answers
523 views

Technical issue in the approach to Lie groups taken in Brian C. Hall's book

I'm teaching Lie groups and Lie Algebras out of Brian C. Hall's book, which I've enjoyed using. I'm confused about a technical hitch though that I'm not sure how to avoid. The approach taken in this ...
6
votes
1answer
392 views

Are the distributive permutation groups linearly primitive?

An action of a group $G$ on a set $X \neq \emptyset$ is called transitive if $\forall x,y \in X$, $\exists g \in G$ such that $g.x = y$. It is called primitive if it is transitive and preserves no ...
6
votes
1answer
194 views

How do I determine a real matrix form for a group representation?

Hello mathoverflow community, I am a little stucked working on my master thesis. For a representation on $\mathbb{Z}_p\ltimes\mathbb{Z}_p^*$ induced from the additive character $\chi$ of ...
4
votes
1answer
104 views

Bott-Samelson construction of a perfect Morse function on G/T

An undergraduate student of mine is interested in writing a "senior" thesis on the topology of Lie groups. Let $G$ be a simply connected compact Lie group and $T$ a maximal torus. One way of ...
9
votes
1answer
230 views

Invariant ring of $S_5$

The irreducible representations of the Symmetric group $S_5$ are classified by the partitions of $5$. For the standard representation which corresponds to the partition (4,1) the ring of invariants is ...
7
votes
1answer
584 views

A Zero-Multiplicity Problem Related to Foulkes' Conjecture

I'm a combinatorialist that is interested in estimating multiplicities of irreps of $1^{S_{kn}}_{S_k \wr S_n}$ (the action of symmetric group on uniform partitions). I'm aware of the difficulty (or ...
7
votes
2answers
201 views

What kind of geometric object is the Pauli spin matrix vector $\vec{\sigma} = (\sigma_1, \sigma_2, \sigma_3)$?

Physicists routinely wrote all 3 Pauli spin matrices as a vector. $$ \sigma_1 = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0\end{array} \right) \hspace{0.25in} \sigma_2 = \left( ...
1
vote
0answers
42 views

Extra-Lorentzian Kac-Moody algebras

My question is about Kac-Moody (KM) algebras of finite rank with symmetrized Cartan matrices $B = C A$ ($A$ is Cartan matrix) of signatures $(-,-,+,...,+)$, $(-,-,-,+,...,+)$, etc. i.e. with ...
0
votes
0answers
69 views

the linear span of all matrix coefficients is $C(G,\mathbb{C})$ where $G$ is a finite group

Theorem. Let $\{(R_{\alpha},V_{\alpha})\}$ be a complete set of inequivalent irreducible finite dimensional representations of a finite group $G$. Let $V_{R_{\alpha}}$ be the subspace generated by all ...
3
votes
0answers
51 views

Computing Springer action on the homology of affine Springer fibers

Lusztig defined (in Sec. 5, also Sage) a Springer action of the affine Weyl group on the homology of affine Springer fibers (Iwahori one, i.e. in an affine flag variety). In the regular semisimple ...
11
votes
1answer
405 views

Plugging $1-x$ into Schur polynomials

I have a symmetric Laurent polynomial $f$ in $k$ variables expressed as a linear combination of Schur polynomials. I'd like to know what happens when I make the substitution $p(x_1,\ldots,x_k)\mapsto ...
1
vote
2answers
228 views

Fundamental invariants for root subsystems

Let $\Phi$ be an irreducible root system of rank $\ell$. The fundamental invariants of $\Phi$ is a set of $\ell$ integers $d_1, \cdots, d_\ell$ canonically attached to $\Phi$. Now suppose $\Psi$ is ...
6
votes
0answers
123 views

An analogue of Deligne--Lusztig theory for real groups?

I am considering the following analogue of Deligne--Lusztig theory: Take e.g. $G=GL_n(\mathbb{C})$, and let $F$ be the complex conjugate, then we have $G^F=GL_n(\mathbb{R})$. Consider the ``Lang ...
6
votes
1answer
158 views

Wavefront sets of irreducible representations with non-integral infinitesimal characters

Let $G$ be a complex reductive algebraic group (connected, simply connected etc), viewed as a real group. We study the representations of $G$, and we follow the notations in the paper of Barbasch and ...
2
votes
1answer
100 views

Maximal possible dimension of abelian Lie subalgebra of Heisenberg Lie algebra of dimension $2n+1$? [closed]

Fix $n \in \mathbb{N}$, and let $\mathfrak{h}_n$ denote the Heisenberg Lie algebra of dimension $2n+1$ (over any given field $k$). Namely, $\mathfrak{h}_n$ is the Lie algebra with basis $x_1, \dots, ...
0
votes
0answers
45 views

How to compute $t_0$ and $r^0$ in Belavin-Drinfeld's classification of solutions of classical Yang-Baxter equations?

I tried to understand Belavin-Drinfeld's classification of solutions of classical Yang-Baxter equations. In the book a guide to quantum groups, on page 83, there is an example of solutions of the ...
3
votes
1answer
116 views

Must normalizing field outer automorphisms “divide” the dimension?

Imprecise question: To get a normalizing field outer automorphism of order $r$, must we multiply the dimension by $r$? Precise hypothesis: Let $p\geqslant 5$ be a prime, let $q$ be a power of $p$ and ...
3
votes
1answer
81 views

Weingarten function for unitary group

Studying integration over unitary group I came across this function, the Weingarten function Wg, such that $$ \int_{\mathcal{U}(N)} \prod_{k=1}^{n} U_{i_kj_k} U^*_{m_k r_k} dU=\sum_{\tau,\sigma\in ...
4
votes
1answer
401 views

What are local spaces and what are they good for?

Factorization structures have been popular in the past decade. Recently a variant of this structure has been suggested by Ivan Mirkovic (and possibly collaborators). This variant, which goes under the ...
12
votes
2answers
395 views

Is the Steinberg representation always irreducible?

Let $\mathbb{F}$ be a field. The Tits building for $\text{SL}_n(\mathbb{F})$, denoted $T_n(\mathbb{F})$, is the simplicial complex whose $k$-simplices are flags $$0 \subsetneq V_0 \subsetneq \cdots ...