Tagged Questions
Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
1
vote
0answers
42 views
Local factors determine Weil representations - proof of the Artin representation case
This post can be seen as a continuation of this post I created on MathOverflow.
I want to understand the proof of the following Theorem from "Euler Factors determine Weil Representations" by Tim and ...
3
votes
0answers
39 views
HKR generalized character theory question regarding a certain construction
In that paper https://web.math.rochester.edu/people/faculty/doug/mypapers/hkr.pdf Hopkins-Kuhn-Ravenel introduced the idea of generalized character corresponding to a complex-oriented cohomology ...
3
votes
1answer
135 views
When is $\Omega^1$ an equivalence?
Let $C$ be an abelian category with enough projectives and $\underline{C}$ the stable category of $C$ that is obtained by factoring out projective modules.
When is the functor $\Omega^1 : \underline{...
1
vote
0answers
47 views
about weight decomposition of U(sl3) [on hold]
the universal enveloping algebra is $\mathbb{Z}^2$-graded algebra. $U(sl_3)= \bigoplus_{(i_1,i_2) \in \mathbb{Z}^2 } U_{i_1\alpha_1+i_2\alpha_2}$ $($the weight decomposition$)$ where $\alpha_1$ and $...
5
votes
0answers
37 views
Cluster-tilting object for a local non-selfinjective algebra
Let $A$ be a non-selfinjective (which is equivalent to non-Gorenstein) local finite dimensional algebra.
Is there a known example of such an $A$ having a cluster-tilting object?
Id be surprised to ...
5
votes
0answers
48 views
Structure of invariant lattices and reductions of group representations with $\text{dim}>2$
Let $G$ be a finite group, $K$ be a $p$-adic field with an uniformizer $\pi$ and residue field $k \cong \Bbb F_q$, and $V$ be an irreducible representation of $G$ over $K$. Consider $X_{V}^G$ the set ...
5
votes
0answers
63 views
Duality between coalgebras and (pseudocompact) algebras - uniqueness
The following result is well-known. It can for example be found in [Iovanov: The representation theory of profinite algebras, Theorem 1.0.2]. For definitions, see below.
Let $k$ be a field. The ...
2
votes
0answers
24 views
Convergence of the intertwining operator as a vector valued integral
Let $G$ be a connected, reductive group over a $p$-adic field with parabolic $P = MN$ defined by a set of simple roots $\theta \subset \Delta$. For $(\pi,V)$ a representation of $M$, and $\nu \in \...
2
votes
0answers
31 views
Projective dimensions of the terms in a minimal injective resolution of the regular module
Let $A$ be a finite dimensional algebra with finite global dimension and with minimal injective coresolution $I_i$ of the regular module $A$.
The study of the projective dimensions of the $I_i$ is an ...
4
votes
0answers
80 views
Number of hereditary modules of a hereditary algebra
Let $Q$ always denote a Dynkin quiver.
Given a connected path algebra $A=kQ$ and a module $M$, is there a useful criterion on $M$ when $End_A(M)$ is again a connected quiver algebra?
Call a module ...
3
votes
1answer
159 views
Are there Galois representations associated with any regular algebraic cuspidal automorphic representation?
Let $K$ be a number field and $\pi$ be a regular algebraic cuspidal automorphic representation on $\mathrm{GL}_2(\mathbb{A}_K)$. Let $\lambda$ be a prime of the field of Fourier coefficients of $\pi$ ...
5
votes
2answers
149 views
Definition of unitary representation of $\mathbf G(\mathbb A_k)$
Let $k$ be a global field, and let $G = \mathbf G(\mathbb A_k)$ for a connected, reductive group $\mathbf G$ over $k$. In these notes by Jayce Getz and Heekyoung Hahn, a unitary representation of $G$ ...
0
votes
0answers
76 views
Gelfand-Pettis integral: what does it mean for a topological vector space to “admit a dual space?”
I am trying to understand more about the Gelfand-Pettis integral. From wikipedia:
What does it mean that $V$ "admits a dual space?" When $V$ is a Banach space, $V^{\ast}$ is taken to be the space ...
17
votes
0answers
304 views
Does $\mathrm{SL}_{n}(\mathbb{Z}/p^{2})$ have the same number of conjugacy classes as $\mathrm{SL}_{n}(\mathbb{F}_{p}[t]/t^{2})$?
Let $p$ be a prime; $\mathbb{F}_{p}$ is the field with $p$ elements
and $\mathbb{F}_{p}[t]$ the ring of polynomials in $t$ over $\mathbb{F}_{p}$.
Does $\mathrm{SL}_{n}(\mathbb{Z}/p^{2})$ have the ...
3
votes
2answers
181 views
Casimir operator of a given Lie algebra and relation with its matrix representation
I'm following Gilmore's recipe to compute the abstract Casimir operator of a given algebra (in this example, I refer to algebra su(2)). This recipe bring up a matrix representation of the algebra and ...
0
votes
0answers
38 views
Localizing a Clebsch-Gordan expansion around one representation
For the Lie group $\mathrm{SU}(2)$ the irreducible representations $\pi_m$ are labelled by non-negative integers $m$ and have dimension $(m+1)$. By the Peter-Weyl theorem, they form a basis for $L^2(\...
1
vote
0answers
65 views
Dimension sum “rules” in Lie algebras
tr;dr intro: I came up with this question when I couldn't remember how many terms are in an $E_7$-ish (representing $\bigotimes$ adjoint) clebsch. Tried it on $G_2$, $7 \bigotimes 14=7+...$ argh, is ...
5
votes
1answer
89 views
SYT and contents of a partition
Let $\lambda$ be an integer partition, denote the number of Standard Young Tableaux of shape $\lambda$ by $f_{\lambda}$. This number is computed by the formula
$$f_{\lambda}=\frac{n!}{\prod_{u\in\...
5
votes
1answer
59 views
Derived equivalences of Artin algebras with finitistic dimension zero
Let $A$ be an Artin algebra of finitistic dimension zero and $B$ an algebra derived equivalent to $A$. Does $B$ also have finitistic dimension zero?
In case this is true, this might generalise the ...
7
votes
1answer
135 views
How should I think about the module of coinvariants of a $G$-module?
Let $G$ be a group, $M$ a $G$-module, then the group of coinvariants is the module $M_G := M/I_GM$, where $I_G$ is the kernel of the augmentation map $\epsilon : \mathbb{Z}G\rightarrow \mathbb{Z}$.
...
2
votes
0answers
55 views
How are the unit/counit of a Hopf algebra and of an categorical adjunction related?
For categories $\mathcal{C}$ and $\mathcal{D}$, a pair of functors $\mathcal{C} \xrightarrow{\;L\;} \mathcal{D}$ and $\mathcal{C} \xleftarrow{\;R\;} \mathcal{D}\,$ are an adjoint pair if we have ...
2
votes
0answers
29 views
Intuition for the Euler form in a finitary category
Suppose that $\mathcal{C}$ is a finitary category, so for any two objects $A$ and $B$ we have that $|\mathrm{Ext}^i(A,B)| < \infty$ for $i\geq 0$, suppose $\mathcal{C}$ has finite global dimension, ...
1
vote
0answers
24 views
Piecewise hereditary algebras of Dynkin type that are QF-3
Is there an easy classification of piecewise hereditary (which means derived equivalent to a hereditary algebra) algebras of Dynkin type
that are Quasi-Frobenius-3 (meaning that the injective envelope ...
4
votes
1answer
65 views
Corollary for Casselman-Shalika formula
Assume $\pi$ is an unramified representation of $GL_n(F)$, where $F$ is a p-adic field. And $\phi$ is an unramified vector for $\pi$. Assume $W_{\phi}$ is a Whittaker function associated to $\phi$. ...
3
votes
1answer
178 views
tensor product of massless Poincare representations
Consider two massless representations of the connected Poincare group $ISO_0(1,3)$ with helicities $s$ and $t$. What is the decomposition of their tensor product into irreducibles?
Massless ...
1
vote
0answers
41 views
A convergence condition on tempered representation
Assume $\pi$ is a tempered representation of $GL_n(\mathbb{Q}_p)$. $N_n$ is a maximal unipotent subgroup of $GL_n$, and $\xi$ is a non-degenerate character of $N_n(\mathbb{Q}_p)$. Let $\Pi$ be the ...
14
votes
0answers
172 views
Young's natural representation of the symmetric group
The literature on the representation theory of the symmetric group contains some terminology that I find puzzling, and I am wondering if someone here knows the full story.
One of the standard ways to ...
4
votes
0answers
58 views
Sum of all projective dimensions of simple modules
Let $X_{n,t}$ be the set of all finite dimensional algebras (we can assume they are given by a connected quiver and admissible relations) that have global dimension equal to $n$ and $t$ simple modules....
8
votes
1answer
119 views
Central extensions of loop groups
Let $LG=\operatorname{Maps}(S^1,G)$ be the loop group of a compact Lie group $G$. I should add some adjectives to $G$, but for sake of simplicity let's just take $G=SU(2)$.
There is a central ...
31
votes
0answers
303 views
Chern character of a Representation
Let $G$ be a finite group. Under the identification of the representation ring $R_{\mathbb{C}}(G)$ with the equivariant K-theory $KU^0_G(\ast)$ of the point, followed by Atiyah-Segal completion-...
3
votes
0answers
160 views
Representations of GL(n,2) over a field of characteristic 2
I would appreciate very much if you can point to me some references on the following:
1) Representations of the linear group $GL(n,2)$ over $F_2$.
2) Representations of $GL(n,2)$ over an algebraic ...
5
votes
1answer
112 views
Is there a subgroup of dual depth 3?
This post is motivated by an exchange with Zhengwei Liu. It is more than the dual version of this post, because we consider any subgroup (instead of just maximal), and even more at the end...
Let's ...
9
votes
2answers
370 views
Closed formulas for the character of the symmetric group
I know the Murnaghan–Nakayama rule, but I am wondering if there is any closed formulas for the character of the symmetric group. I know the following:
$$\chi_{n}(\sigma) = 1$$
$$\chi_{11...1}(\sigma) ...
4
votes
0answers
114 views
Question on syzygies
Given a finite dimensional algebra $A$ over a field $K$ with $\Omega^i(D(A)) \cong \Omega^{i+1}(D(A))$ for some $i \geq 1$, where $D(A)=\operatorname{Hom}_K(A,K)$.
Do we then also have $\Omega^{-i}(A)...
5
votes
1answer
333 views
Big finitistic dimension of finite dimensional algebra
Let $A$ be a finite dimensional algebra. The big finitistic dimension of $A$
is
$$\operatorname{FinDim}(A)=\sup\{\operatorname{pd}(M)\mid M\in \text{Mod-}A \text{ and } \operatorname{pd}(M)<\...
3
votes
0answers
86 views
A Question on Plancherel decomposition of p-adic groups
Assume $G$ is a $p$-adic group. If $\pi$ is an irreducible quotient of $C_c^\infty(G)$, then there is a surjection $C_c^\infty(G)\twoheadrightarrow \operatorname{Hom}_G(C_c^\infty(G),\pi)^\star\otimes\...
6
votes
3answers
287 views
Is there a maximal subgroup of depth 3?
Let's first define what we mean by depth of a subgroup.
Let $G$ be a finite group and $H$ a subgroup. Let $(V_i)_{i \in I}$ and $(W_j)_{j \in J}$ be the irreducible complex representations of $G$ ...
3
votes
1answer
137 views
Eisenstein series for discrete subgroups of SL(2,C)?
I am looking for a reference for Eisenstein series for discrete subgroups of $SL(2,\mathbb C)$, in particular, finite index subgroups of $SL(2,\mathcal O_K)$ where $K$ is an imaginary quadratic field.
...
2
votes
1answer
100 views
$L^2(X) \cong L^2(X',\xi)$
Recently, I read a notes about Sakellaridis and Venkatesh conjecture. It mentions a technique called "unfolding" and gives an example:
Let X=A\G, X'=N\G, where G=PGL(2), A={
$\left[\begin{array}{...
2
votes
1answer
167 views
Local factors determine Weil representations - proof of the cyclic case
I already created this post on Math Stack Exchange but I was not so sure if this question fits better here. If it is not, I want to apologize in advance and feel free to delete my post.
I want to ...
1
vote
1answer
61 views
$\Lambda$ has infinite repetition index, rep$(\Lambda /J)=\infty$, why?
https://arxiv.org/pdf/1407.2321v1.pdf $P_{6}$, Example 2.3(c) If $\Lambda=K[X^{2}, Y^{2}]/(X^{2}, Y^{2})$ for a field,
then $\Lambda$ has infinite repetition index and, a fortiori, infinite syzygy ...
9
votes
1answer
143 views
Representations of the automorphism group of graphs via spectral graphs theory
Given a (simple) graph $G=(V,E)$ with $V=\{1,...,n\}$ and let $A$ be its adjacency matrix.
I am interested in the representation theory (over $\Bbb R$) of the automorphism group $\def\Aut{\mathrm{Aut}...
3
votes
0answers
102 views
Two notions of a “nilpotent orbit”
I am wondering about the equivalence of two notions of a "nilpotent orbit".
The first notion, which I am familiar with, is as follows: given a lie group $G$ and a lie algebra $\frak{g}$, the orbit of ...
5
votes
0answers
164 views
Projecting GxG onto subspace with tied irreducible representations
Suppose I have a finite group $G$. With this group, I can associate an ortho-normal Hilbert space spanned by elements of the group $$\mathcal{H} = \{|g\rangle: g \in G \}$$.
I could alternatively ...
5
votes
0answers
112 views
Tannakian theory for Lie algebras
Let $G$ be a reductive (just in case) linear algebraic group over $\mathbb{C}$ and let $\mathfrak{g}$ be the Lie algebra of $G$. Consider the category $\operatorname{Rep}(G)$ of finite dimensional ...
2
votes
0answers
40 views
Hecke algebra and closed subgroup
As we know for a locally compact group $G$, given an open compact subgroup $K$ and an irreducible representation $\rho$ of $K$, there is an equivalence of categories between the irreducible ...
16
votes
0answers
146 views
Limiting representation theory of quantum groups at roots of unity and SL(2,C)
Let $V_N$ denote the $N$-dimensional representation of the quantum group $U_q(\mathfrak s\mathfrak l_2)$. I am told that in the limit $N\to\infty$ with $q=e^{2\pi i/n}$ and $N/n\to\alpha\in(0,1)$, ...
4
votes
1answer
145 views
The embedding of $\mathfrak{g}_2$ into $\mathfrak{b}_3$ ($\mathfrak{so}_7$) on Chevalley generators
Let $\mathfrak{g}_2$ / $\mathfrak{b}_3$ be the simple complex Lie algebra of type $\mathsf{G}_2$ / $\mathsf{B}_3$ (the latter is also known as $\mathfrak{so}_7$).
How is the embedding $\mathfrak{g}...
3
votes
0answers
91 views
Can a acyclic quiver algebra be derived equivalent to a non-acyclic quiver algebra?
Can a quiver algebra with acyclic quiver be derived equivalent to a quiver algebra with non-acyclic quiver?
(I moved this question from another thread Derived equivalences of Dyck paths , where the ...
4
votes
0answers
59 views
semisimplicity of maps in braided vector spaces
Let $V$ be a finite dimensional braided vector space over $\mathbb{C}$.
This means that we have a map $$c_{V,V}:V\otimes V\to V\otimes V$$ which gives us an action of the braid group $B_n$ on $V^{\...
