0
votes
1answer
41 views

Does Schur's Lemma hold in this case? Regular representations of $S_n$ over $\mathbb R$

Warning I am a physicist and I am not familiar with a lot of the machinary of representation theory. I consider the regular representation of $\mathbb S_n$ over reals $\mathbb R$ ($\mathbb R \mathbb ...
0
votes
0answers
12 views

Projective dimension of a sub-ideal

Let $\mathbf{k}$ be a field, and let $S=\mathbf{k}[x_1,x_2,\ldots,x_n]$. Let $I\subset J$ be finitely generated monomial ideals in $S$. Is it true that the projective dimension of $I$ is either ...
-4
votes
0answers
23 views

How to know which make which surjective and which Tor is correct to represent them surjective for codordism [on hold]

when A connect B through cobordism A --- cobordism ----B from view of function, when define surjective there exist a function g to make f surjective such that g ...
2
votes
1answer
79 views

Various definitions of the Bruhat decomposition of the affine Grassmannian

Let $G$ be a connected, simply connected, complex, semi-simple group with affine Grassmannian $\mathcal Gr \cong G(\mathbb C[t,t^{-1}])/G(\mathbb C[t])$. Fix a choice of maximal torus $T \subset G$ ...
0
votes
0answers
43 views

Generalized weight space

In their paper Lepowsky and Mcmollum sketch theory of weights in a more general setting. Here is their definition of a weight space: If $A$ is a subset of $\mathfrak g$ and $\lambda$ is a function ...
0
votes
0answers
24 views

question about Baer sum of extensions

Let $E_1$ and $E_2$ be extensions of $\mu_p$ by $\mathbb{Z}/{p\mathbb{Z}}$. Assume that $E$ contains $E_1$ and $E_2$ both, and $E_1 \cap E_2 = \mathbb{Z}/{p\mathbb{Z}}$. Then, does $E$ contain their ...
1
vote
0answers
29 views

Classification properties of fusion rings

Fusion rings have so many classification properties (I checked the literature a bit) that my head hurts. For practical reasons I define the following three new properties (which might coincide with ...
1
vote
0answers
27 views

Anti-Invariant Polynomials of the Dihedral group

I'm interested in the one-dimensional irreducible representations of $D_{2n}$ acting on $\mathbb{R}[x,y]$. I have found that the trivial representations for an algebra freely generated by $x^2+y^2$ ...
2
votes
0answers
49 views

Invariant Theory over finite adeles

Classical invariant theory, among the other things, classifies polynomial functions over a vector space $V$ endowed with a quadratic form $Q$ which are invariant under the action of $SO(V,Q)$. I am ...
0
votes
1answer
107 views

Which $\frak{sl}_2$-Representations Arise From Hermitian Metrics

Recal that $\frak{sl}_2$ is the Lie algebra with basis elements $e,f,h$, and bracket $$ [e,f] = h, ~~~ [h,e] = 2e, ~~~ [h,f] = -2f. $$ For $M$ a $2n$-complex manifold, the Lefschetz identities tell us ...
1
vote
0answers
40 views

Projection formula for smooth representations of locally profinite groups

Let $G$ be a locally profinite (i.e., locally compact, Hausdorff, and totally disconnected) topological group, $H \le G$ a closed subgroup, $(\pi, V)$ a smooth representation of $G$, and $(\sigma, W)$ ...
1
vote
0answers
56 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 ...
1
vote
0answers
56 views

Homology and Exterior Square

Let $G$ be a finite group and $G \wedge G$ denote the exterior square of $G$. It is well known that the second integral homology $H_2(G,\mathbb{Z})$ is the kernel of homomorphism $x \wedge y \mapsto ...
3
votes
4answers
321 views

Representations of the two dimensional non-abelian Lie algebra

I would like to know a complete description of the indecomposable representations of the two dimensional non-abelian Lie algebra over the complex numbers. The finite dimensional representations would ...
0
votes
1answer
199 views

A functorial isomorphism in derived category

This question is a direct continuation of Question 1 in this post: Two basic questions on derived categories Let $f\colon \mathcal{A}\to\mathcal{B}$ be a left exact functor between two abelian ...
4
votes
1answer
139 views

Shift-invariant symmetric functions in representation theory?

The connection between symmetric functions and representation theory is well-known. Now consider the subspace of symmetric functions that are shift-invariant, that is, functions satisfying ...
4
votes
3answers
248 views

Poincare duality for (co)homology of Lie algebras?

Let $R$ be a commutative ring and $\mathfrak{g}$ a Lie $R$-algebra that has an $R$-module basis with $n$ elements. In Algebra, Geometry, and Software Systems by Joswig & Takayama on p.200 it ...
7
votes
1answer
164 views

On unramified p-adic groups

Let G be a reductive group over a local field F. Let O be the ring of integers of F. The following are equivalent (and groups satisfying these conditions are called unramified): (a) G is quasisplit ...
-1
votes
1answer
79 views

terminology: “complex” and “sequence” in homological algebra

It appears that the terms "complex" and "sequence" are used synonymously in homological algebra. But there seem to be collocations (in the linguistic sense) that prefer one of those words. For ...
2
votes
0answers
88 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 ...
2
votes
1answer
72 views

Weight polytopes of the fundamental representations of simple Lie groups

Where can I find a description of the weight polytopes of the fundamental representations of the classical complex simple Lie groups? Thanks in advance
4
votes
0answers
157 views
+50

Decompositions of a compact Lie group into “fixed point set types”

Consider a compact Lie group $G$ which acts on a closed Riemannian manifold $M$ by isometries. Then it is well-known that there are only finitely many isotropy types of the $G$-action, i.e. finitely ...
-5
votes
0answers
90 views

Could it be possible to use Selberg trace formula to prove that the irreducible characters of representation form an orthonormal basis? [closed]

G is a finite group.H is the space of class functions on G;the irreducible characters χ1,χ2...χn belong to H form an orthonormal basis. That is an basic representation theorem,and you can easily prove ...
7
votes
0answers
112 views

What's the appropriate notion of a Unitary representation of a Lie algebra?

Here Lie algebras/groups are real. The most straightforward definition might be: Def: A representation $\rho:\mathfrak{g} \rightarrow \mathfrak{gl}(V)$ is unitary if $V$ is equipped with a Hermitian ...
1
vote
1answer
133 views

Homomorphisms from irreducible spaces to reducible spaces

Let $P_{\lambda}$ be a Young symmetriser associated to the following tableau $(a_1 a_2 a_3 b_3 ; b_1 b_2)$ where the entries seperated by the ; belong to first and second COLUMNS of the tableau. Take ...
2
votes
0answers
53 views

relations between derived categories of ind-A and A

Let $A$ be an abelian category and $indA$ be its ind category. I want to know the relations between $D^b(A)$ and $D^b(indA)$. For example, I find in another question that if $A$ is thick in $indA$, ...
3
votes
0answers
101 views

Unipotent representations of SL(2,R) by quantization

I'm a PhD student in mathematical physics and I happen to need some elements of Kirillov's "orbit method" for producing representations of Lie groups. I'm familiar with symplectic geometry, geometric ...
3
votes
2answers
504 views

Two basic questions on derived categories

Let $\mathcal{A}, \mathcal{B}$ be two abelian categories with sufficiently many injective objects (in my case these are categories of sheaves of vector spaces on a manifold). Let $f_*\colon ...
-1
votes
0answers
90 views

Correct definition of a Gerstenhaber algebra in the DG setting

(Edit: Nevermind the question. I think I know where my misunderstanding lies now.) It might seem like a stupid question, but let me argue that it is not. The first reaction that one might have ...
7
votes
2answers
115 views

Can monomial representations induced from nonmonomial representations?

Let $H$ be a subgroup of $G$. Let $\rho$ be an irr representation of $G$ induced from an irr representation $\theta$ of $H$. It is well known that $\rho$ is monomial if $\theta$ is monomial. Is it ...
1
vote
0answers
107 views

Non invertibility of certain integral arising from group action

Let a compact topological group $G$ with invariant measure $\mu,$ acts on a simply connected compact topological space $X$ and $\rho$ is a $n$-dimensional unitary representation of $G$. ...
2
votes
1answer
73 views

No basis change in a fusion ring allowed?

Consider the Fibonacci category F: $A\bigotimes{A}=E\bigoplus{A}$. A study by Study (SCNR :-) already 1890 listed all unital associative algebras (with rank<=4). ...
0
votes
0answers
47 views

Yoneda extension in the category of representations

Assume $G$ is a group scheme over a field $k$ and consider the categories $Rep_G$ of finite dimensional representations of G and $REP_G$ of all representations of G. For two objects $A,B$ in $Rep_G$, ...
5
votes
2answers
168 views

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 $l$ ...
2
votes
1answer
90 views

A quadratic algebra with four generators and four relations

Algebra that I'm going to describe pop-up in my research, it looks completely elementary, but I don't know any appropriate references. Let $k$ be an algebraically closed field of characteristic ...
2
votes
0answers
61 views

A question on Plancherel measure for $p$-adic group

Let $G$ be a reductive group over a $p$-adic field. My understanding is that the Plancherel measure on $G$ is a measure on the unitary dual $\hat{G}$. But at the same time, for example, in his famous ...
0
votes
0answers
74 views

G-equivariant coherent sheaves on Bott-Samelson Resolutions

Let $G$ be a Lie group, $B$ be a Borel subgroup. $G/B$ is the corresponding flag variety. Let $w$ be an element of the Weyl group $W$ with a reduced expression $w = s_1 \cdots s_n$. Let $X_w$ be ...
4
votes
2answers
132 views

Weight multiplicity formulae for $(\mathfrak g,B)$-irreps

Let $G$ be a complex reductive Lie group, $B$ a Borel subgroup, with which to define "dominant weight". Let $\lambda$ be an integral weight, not necessarily dominant, but nonetheless giving a ...
13
votes
2answers
662 views

Is homology finitely generated as an algebra?

If a differential graded algebra is finitely generated as an algebra, is its homology finitely generated as an algebra? Is it easier if we impose any of the three conditions: characteristic zero; ...
2
votes
2answers
150 views

Standard homology result on double complexes

Suppose you have got a double complex in an abelian category with objects $(A_{rs},d_{rs})$ such that $A_{rs} = 0$ for $r < 0$ or $s < 0$. Suppose furthermore that the rows ...
1
vote
1answer
151 views

When is the Ad (Adjoint Representation) Morphism a Closed Map

Given a Lie group $\mathfrak{G}$ with finite centre and with Lie algebra $\mathfrak{g}$, I am looking at a simple proof that negative definite Killing form implies compactness. This proof is given ...
2
votes
1answer
97 views

Number of double cosets of a Young subgroup

Let $\lambda\vdash n$ be a partition of $n$ with $k$ parts and $S_\lambda$ be a Young subgroup of $S_n$. Further let $S_\lambda\backslash S_n/ S_\lambda$ be the set of double-cosets. Now I would like ...
3
votes
1answer
141 views

Restricting the Steinberg representation of $SL_{2n}$ over a finite field to the symplectic group

Let $\text{St}_n(\mathbb{F}_q)$ be the Steinberg module (over $\mathbb{C}$) for $\text{SL}_n(\mathbb{F}_q)$. What is the irreducible decomposition of the restriction of $\text{St}_{2n}(\mathbb{F}_q)$ ...
0
votes
0answers
31 views

Corestriction map for the relative homology (cohomology) group

Let $G$ be a group and $N$ be its normal subgroup. Is there any concept of corestriction map for the relative homology (cohomology) group $H_n(G,N,-)$ ($H^n(G,N,-)$) such that when $N=G$ it is the ...
2
votes
1answer
121 views

Tangent space of the Fourier algebra $A(G)$

Let $G$ be a real Lie group and $A(G)$ be its Fourier algebra. Let us call a linear continuous functional $f:A(G)\to{\mathbb C}$ a tangent vector of $A(G)$ in the point $a\in G$, if it satisfies the ...
10
votes
0answers
182 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 ...
0
votes
0answers
35 views

a generalization of the annihilator of cokernel ideal

Let $R$ be a (commutative, associative, unital) ring, consider a homomorphism of some (finitely generated) free $R$-modules $F\stackrel{A}{\rightarrow}G$. Its basic invariants are the Fitting ideals, ...
3
votes
0answers
119 views

What do we know about isospectral Cayley graphs?

Let $\Gamma_1=\Gamma_1(G_1,S_1)$ (respectively $\Gamma_2=\Gamma_2(G_2,S_2)$) be a Cayley graph for the group $G_1$ and the finite generating set $S_1$ (respectively $G_2$ and the finite generating set ...
0
votes
0answers
107 views

Irreducible action of an algebraic group

Is the following claim true?: Let $G$ be an algebraic group such that $G^\circ$ is reductive. Suppose $G$ acts irreducibly on $V$. Is it true that $V$ is decomposed (written as direct sum) into ...
2
votes
1answer
112 views

When is the Fourier algebra $A(G)$ enough close to the Fourier-Stieltjes algebra $B(G)$?

I am reading P.Eymard's paper on the Fourier algebras of locally compact groups, and I have several questions about his constructions. I asked one of them in math.stackexchange, so far without ...