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

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5
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114 views

Intersections of the B-orbits and the orbits of some other Borel subgroups in the flag variety G/B

This is a follow-up of this previous question below: Intersections of $B$ and $B^-$ orbits in the flag variety $G/B$ Let $G = SL_n(\mathbb{C})$, $B$ be the standard Borel subgroup, and consider some ...
5
votes
0answers
80 views

How to characterize the class of $(\mathfrak{g},K)$-modules with a fixed lowest K-type in the framework of D-modules?

Let $G$ be a real semisimple Lie group, $K$ be a maximal compact subgroup. Let $\mathfrak{g}_0$ and $\mathfrak{k}_0$ be their real Lie algebras respectively. Let $\mathfrak{g}$ and $\mathfrak{k}$ be ...
0
votes
0answers
107 views

Anticommuting operators with positive properties

Which classes of $M\in \mathsf M_k(\Bbb R)^{n\times n}$ admit solutions $N\in \mathsf M_k(\Bbb R)^{n\times n}$ such that $$(M\otimes N+N\otimes M)(u\otimes u)=0$$ forall $u\in \mathsf D_k(\Bbb ...
2
votes
0answers
97 views

Adding a row to a Young Tableau via Novelli-Pak-Stoyanovskii

Let $T_{\lambda}$ be the set of standard young tableaux (SYT) of shape $\lambda_1\geq \lambda_2\cdots\geq \lambda_n$. Now consider pushing a row $\mu$ with $\mu\geq \lambda_1$ onto $Y$ to give shape ...
5
votes
0answers
111 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
327 views

Decomposing a reducible representation of the unitary group

Consider the representation $L_U$ of the unitary group $U(n)$ on $L(\mathbb{C}^n)$ where $L_U$: $L(\mathbb{C}^n) \rightarrow L(\mathbb{C}^n)$ is a linear operator that $L_U M=U M U^{\dagger} $, ...
2
votes
0answers
104 views

What's the relation of the Hecke algebra of a pair and the flag variety?

Let $G$ be a real semisimple Lie group and $K$ a maximal compact subgroup. Let $\mathfrak{g}$ and $\mathfrak{k}$ be the complexified Lie algebra of $G$ and $K$, respectively. Then the Hecke algebra ...
5
votes
2answers
330 views

Concise mathematical definition of the fusion product on the Verlinde ring?

The Verlinde ring of a (let us say) simply connected simple compact Lie group has as underlying additive group the Grothendieck group of representations of the central extension $\widehat{LG}$ of the ...
1
vote
1answer
84 views

Explicit deformations of pseudo representations

Let $G$ be a group (which I will be glad to consider to be the absolute Galois group of a $p$-adic field, and so satisfies Mazur's finiteness condition which appears in his paper Deforming Galois ...
2
votes
0answers
139 views

Free-field representations: how to study highest-weight submodules of the Fock module?

Suppose we have a representation of some affine Lie algebra $\mathfrak{g}=\mathfrak{n}_- \oplus \mathfrak{h} \oplus \mathfrak{n}_+$ on a Fock space $V$. The module $V$ will contain a lot of ...
2
votes
1answer
314 views

Canonical representation of $\operatorname{SL}(2,\mathbb{R})$ on $L^2(\mathbb{R}^2)$

As a unimodular subgroup of the group of automorphisms of $\mathbb{R}^2$, $\operatorname{SL}(2,\mathbb{R})$ can be represented as a subgroup of $\mathcal{U}(L^2(\mathbb{R}^2))$ (the group of unitary ...
2
votes
1answer
131 views

cohomology of orthogonal (or general linear) group over finite fields

Let $\mathbb{Z}_2=\mathbb{Z}/2\mathbb{Z}$. Let $$ O(\mathbb{Z}_2^{\oplus k})=\{A\mid A \text{ is a } k\times k \text{ - matrix with entries } 0,1, det(A)=\pm 1\} $$ What is $$ ...
6
votes
1answer
238 views

In a closed monoidal abelian category, are the compact projectives a monoidal subcategory?

Question: In a closed monoidal abelian category such that the unit object is compact projective, must the tensor product of compact projective objects be compact projective? Recall that an object ...
5
votes
0answers
149 views

Counting points on Hessenberg varieties over a finite field

Let $G$ be an connected reductive group over finite field $k$. I will assume that $\text{char}(k)$ is very good for $G$ (or even larger, if preferred). Let $B\subset G$ be a Borel subgroup defined ...
2
votes
0answers
61 views

explicit matrices for Weil ($p^2$ dimensional) representation of $Sp(4,\mathbb{F}_p)$, $p>3$

I am looking for more-or-less explicit matrices for the $p^2$ dimensional Weil representation of $Sp(4,\mathbb{F}_p)$, suitable for computer implementation. Ideally, I would like the images of the ...
17
votes
3answers
705 views

Center of a simply-connected simple compact Lie group and McKay correspondence

Let $G$ be a simply-connected simple compact Lie group. Its center $Z(G)$ is a finite abelian group, say $Z(G) = \mathbb Z/k\mathbb Z$ for $G=SU(k)$. I find the following interpretation of $Z(G)$ in ...
3
votes
0answers
71 views

Linear independence of points under projection of Veronese re-embedding

Let $V$ be a complex vector space. Let $x_1,...,x_k\in PV$. Let $v_d: PV\rightarrow PS^dV$ be the Veronese. Then $v_d(x_1),...,v_d(x_k)$ are in general linear position as long as $k\leq d-1$. Now let ...
1
vote
1answer
191 views

cohomology of orthogonal group of integers

Let $$ O(\mathbb{Z}^{\oplus k})=GL(\mathbb{Z}^{\oplus k})\cap O(k). $$ What is $$ H^*(BO(\mathbb{Z}^{\oplus k});\mathbb{Z})? $$ If it cannot be computed out, can we get $$ H^*(O(\mathbb{Z}^{\oplus ...
0
votes
0answers
81 views

On continuous part of the L^2 spectrum

Suppose $G$ is a real reductive Lie group and $\Gamma$ is a lattice in $G$ (of finite co-volume). I am reading Langlands's paper " On the functional equation satisfied by the Eisenstein series". I ...
3
votes
1answer
268 views

Number of semi-standard tableau

What is the number of semi-standard tableau (weakly increasing on rows and strictly increasing on columns) for the partition $2n=n+n$ with entries $\{1,2, \cdots ,n\}$ such that each $i$ appears ...
2
votes
0answers
374 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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0answers
95 views

Central extensions of SL2(R) by U(1) ?

Can somebody please tell me what are the central extensions of SL2(R) by U(1), that is, what is $H^2(SL2(R), U(1)) $ ? Thank you
2
votes
1answer
206 views

Relation between Different Definitions of Induced Representation

I've seen two different ways to define induced representation. One is as in the book Introduction to representation theory: If $G$ is a group, $H$ is a subgroup of it, and $V$ is a representation of ...
2
votes
0answers
90 views

Dimension of affine Springer fiber and its functor of points as an ind-scheme

Let $k$ be a finite field and let $F = k( (t))$ with ring of integers $\mathfrak{o} = k[ [t]]$. Let $G$ be a connected linear algebraic $k$-group with Lie algebra $\mathfrak{g}$. Suppose that ...
6
votes
1answer
264 views

Jordan decomposition of the tensor product of two matrices

I asked this question on Math.SE here, but did not get a lot of attention. I am interested in the problem of determining the Jordan decomposition of the tensor product of two unipotent matrices over ...
8
votes
1answer
347 views

Automorphisms of generic complete intersections

This question concerns a seemingly folk lore result, which states that automorphism groups of generic complete intersections are trivial, under certain assumptions. To state the question, let $r \geq ...
6
votes
0answers
250 views

What is miraculous about the mirabolic subgroup?

I recently asked this question about Euler subgroups and generalizing the automorphic theory of $\mathrm{GL}_n$ to a more general setting. My question here is more specific. As mentioned there, the ...
2
votes
1answer
118 views

Projection and representation

Suppose that $\rho : G \longrightarrow U_n(\mathbb C)$ is an irreducible representation of group $G$. Suppose that $P$ is a projection of $\mathbb C^n$ into a subspace of small codimension (i.e. of ...
3
votes
1answer
183 views

Which group algebras in analysis are “true group algebras”?

Let $G$ be a group, $A$ a unital associative algebra over ${\mathbb C}$, and let us call a representation of $G$ in $A$ an arbitrary map $\pi:G\to A$ such that $$ \pi(1)=1,\qquad \pi(a\cdot ...
1
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0answers
124 views

counting how many boxes from a given Young tableau contribute to hook length made out of two YTs

Think of a Young diagram as a collection of rows with numbers of elements $\mu_1 \geq \mu_2 \geq \cdots \geq \mu_d \geq \mu_{d+1}=0$ (and $\mu_k=0$ for $k>d$) and define for $s=(i,j)$ (where $i$ ...
5
votes
0answers
159 views

Euler Subgroups and Automorphic L-functions

Recently, I have read about the Whittaker expansion for $\mathrm{GL}_n$ and was struck by the utility of the mirabolic subgroup, $\mathrm{P}_n\subset \mathrm{GL}_n$ of matrices with bottom row $(0\; 0 ...
2
votes
2answers
113 views

admissible characters for $PGL_{2}(F)$

What are the irreducible admissible representations of $PGL_{2}(F)$ for $F$ a local nonarchimedean field and do we have formulas for their characters?
2
votes
0answers
280 views

When a Spherical variety is $K$-stable

Let $X=G/H$ for a reductive group $G$ and $X$ is normal and has an open $B$-orbit, for a Borel subgroup $B$ then we call $X$ spherical. My question is When a Spherical variety is $K$-stable? Is ...
3
votes
3answers
441 views

classifying space and cohomology of integer general linear group

I have obtained that the classifying space $$ BGL(\mathbb{R}^n)=BO(\mathbb{R}^n)=G_n(\mathbb{R}^\infty) $$ is the Grassmannian. I have also obtained that the mod 2 cohomology is the polynomial ...
5
votes
0answers
118 views

LS paths construction

Let $W$ be the Weyl group of a simple Lie algebra $\mathfrak L$, and for a dominant weight $\lambda$ denote by $W_{\lambda}$ the stabilizer of $\lambda$ in $W$. Let $\leq$ be the Bruhat order on ...
3
votes
2answers
355 views

symmetric 2-cocycle / many projective representations

Let $G$ be a finite group, $k$ the field of complex numbers. Are there (cohomologically nontrivial) group 2-cocycles $\sigma\in Z^2(G,k^\times)$ such that for all $g,h\in G$: ...
-2
votes
1answer
174 views

Is there any Lefschetz-like principle for representations of finite groups?

Representation theory (at least the origin of this terminology) aims to exhibit a model (a represetative) in the group of matrices for an abstract group which is known by only its group law. So ...
4
votes
1answer
401 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: $\mathcal{L}(H\subset G)$ ...
3
votes
0answers
54 views

Signs associated to self-dual simple objects in a fusion category

Every self-dual simple object $X$ in a fusion category can canonically be assigned a number $a$, from its "snake" associator element: The square of $a$ equals Muger's "squared dimension" of $X$, an ...
0
votes
0answers
115 views

Weyl group representation

Let $G$ be a reductive p-adic group. Let $W$ be a weyl group. if $w$, and $w_o \in W$. I want to know in which case we have $w w_o w^{-1}= w_o$ ? in case if $w_o(\theta)=\theta $ where $\theta$ is a ...
5
votes
0answers
186 views

Conjugation of the quotient of $SL(n,\mathbb{C})$ by a finite subgroup

EDITED Let $G={SL}_{n,{\mathbb{C}}}$, the special linear group over ${\mathbb{C}}$. Let $H\subset G$ be a finite subgroup. Set $X=G/H$ be the corresponding homogeneous space, it is a complex variety. ...
2
votes
1answer
194 views

An expectation of the product of random unitaries

I want to find the answer of $$\int dU \ U^m X \ U^{\dagger m}$$ Where $m\in\mathbb{N}$ and $U$'s are unitary matrices in $U(n)$ and $dU$ is a normalized Haar measure. $X$ is a given self-adjoint ...
2
votes
0answers
45 views

“Prime” fusion rings

Surely this concept is known! (But I don't recall seeing it - maybe under another name? But "prime" is the obvious name choice.) Example. Open the Gepner/Kapustin paper at ...
5
votes
2answers
393 views

Expectation of trace of nth power of unitary matrices

I am trying to find the answer of $$\int dU \ |Tr(U^m)|^2$$ where $m\in\mathbb{N}$ and $U$'s are unitary matrices in $\textit{U}(n)$ and $dU$ is a normalized Haar measure. In the case $m=1$, the ...
19
votes
1answer
480 views

How to make the Capelli's identity less mysterious?

The formulation of the Capelli's identity is very elementary; it has important applications in invariant theory and representation theory, see http://en.wikipedia.org/wiki/Capelli%27s_identity To ...
4
votes
0answers
288 views

An integral with respect to the Haar measure on a unitary group

Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate $$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$ where $dH$ is the unit invariant Haar measure on the group of unitary ...
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176 views

What is the spectrum of $L^1(G:H)$?

Let $H$ be a compact subgroup of a locally compact topological group $G$ and $$ L^1(G:H)=\{f\in L^1(G): R_h f=f\;\text{ a.e. }\; \forall h \in H\}$$ and $\widehat{(G:H)}=\{\xi\in ...
1
vote
1answer
147 views

Isomorphisms of Positive and Negative Spinor Bundles

Here is an extract of the doctoral thesis of C. Lewis under the supervision of D. Joyce (https://people.maths.ox.ac.uk/joyce/theses/LewisDPhil.pdf, 1998): 2.6 Spin Bundles and the Dirac Operator ...
4
votes
1answer
185 views

Continuous-piecewise-linear versus piecewise-linear

Some authors use the term "continuous piecewise-linear" where other authors use the shorter term "piecewise-linear" (with continuity tacit). I'd be interested in people's thoughts about this ...
2
votes
0answers
127 views

Invariant generalized sections of dual vector bundles

Assume X is a real smooth manifold with an action of the real Lie group G. Let E be a G-vector bundle over X. Consider the spaces of generalized sections over X of E, and of E^* (fiberwise dual). My ...