Questions tagged [rt.representation-theory]

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

Filter by
Sorted by
Tagged with
5 votes
0 answers
112 views

Problem with affine root subgroups of $SU_3$ ramified, residue characteristic $p=2$

Let $L/K$ be ramified quadratic extension of local fields, and let characteristic of the residue field of $K$ be $2$. Let $\mathbb{G}=SU_3$, $G=\mathbb{G}(K)$. Let $\text{val}$ be a valuation on $K$ ...
pbarron's user avatar
  • 71
5 votes
0 answers
97 views

Form on symmetric functions and their q,t- analogues

[Notations are as in Macdonald's Symmetric Functions and Hall Polynomials] The space of symmetric functions $\Lambda_{\mathbb{Q}}$ has a bilinear form defined by $ (p_\lambda, p_\mu)= z_\lambda \...
ArB's user avatar
  • 688
5 votes
0 answers
146 views

Quiver and relations of $F\mathrm{SL}(2,q)$

$\DeclareMathOperator\SL{SL}$Let $q=p^n$ be a prime power and $F$ a field of characteristic two. Let $G=SL(2,q)$ the group of $2 \times 2$ special linear matrices over the field with $q$ elements with ...
Mare's user avatar
  • 25.8k
5 votes
0 answers
132 views

A practical way to check whether a module is periodic

A module $M$ over a finite dimensional selfinjective algebra $A$ over a field $K$ is called periodic if $M \cong \Omega^n(M)$ for some $n \geq 1$. We assume here that $M$ is simple and that A is a ...
Mare's user avatar
  • 25.8k
5 votes
0 answers
253 views

Which tensor power of a given representation contains the trivial one?

If $R$ is an irreducible representation of a simple Lie-groups $G$ I assume there is always a lowest integer $n$ such that the tensor product representation $R \otimes R \otimes \ldots \otimes R$ (n ...
Fetchinson0234's user avatar
5 votes
0 answers
347 views

Why admissible representations?

In the theory of automorphic forms we often right away reduce the study to admissible representations, and I wonder how much everything breaks when not. If I understood well, admissible ...
Wirdspan's user avatar
  • 181
5 votes
0 answers
115 views

An intelligent ant living on a symmetric quiver algebra - Does it have a way to find out whether it lives on a trivial extension?

For a given algebra $B$ over a field $K$ the trivial extension $T(B)$ of $B$ is defined as follows: The underlying vectorspace is $T(B)=B \oplus D(B)$ where $D(B)=Hom_K(B,K)$ and the multiplication is ...
Mare's user avatar
  • 25.8k
5 votes
0 answers
76 views

Reference on two numbers associated to a module of finite homological dimension

Let $A$ be a finite dimensional algebra over a field $K$ with a module $M$ which has finite projective dimension and finite injective dimension. Let $n \geq 1$. Let $(P_i)$ be a minimal projective ...
Mare's user avatar
  • 25.8k
5 votes
0 answers
99 views

Decomposition of the Schwartz space as a representation for the orthogonal group

The Schwartz space $\mathcal{S}(\mathbb{R}^n)$ is naturally a $O_n(\mathbb{R})$-representation. I'm assuming that this is a relatively well-behaved representation among the infinite-dimensional ones ...
Johannes Hahn's user avatar
5 votes
0 answers
292 views

Equivalent definitions of unramified characters

Let $G$ be a connected reductive group over a local field $F$. An unramified character of $G(F)$ is a continuous character $\chi: G(F)\to\mathbb{C}^\times$ that is trivial on all compact subgroups of $...
user449595's user avatar
5 votes
0 answers
115 views

Where to read about the toric variety coming from a principal nilpotent element of a (semi)simple algebraic group?

Given a principal (regular) nilpotent element $e$ in the Lie algebra $\mathfrak g$ of a complex semisimple algebraic group $G$, let $\mathfrak s=(e,f,h)$ be an $\mathfrak{sl}_2$-triple for $e$. Then ...
მამუკა ჯიბლაძე's user avatar
5 votes
0 answers
809 views

Quotient of a Lie algebra by a subalgebra - what is it?

Cross-posting from math.SE (asked there 20 days ago). The quotient $G/H$ of a group $G$ by its subgroup $H$ has a $G$-action - every transitive $G$-set is of this form. However, the quotient space $\...
მამუკა ჯიბლაძე's user avatar
5 votes
0 answers
93 views

Periodics of Coxeter matrices for truncated Nakayama algebras

For $n \geq 3$ and $r \geq 3$ let $C_{n,r}=(c_{i,j})$ denote the $n \times n$-matrix where $c_{i,j}=1$ for $j=i,\dots,i+r-1$ (we only do this until $i+r-1>n$). So for example for $n=7$ and $r=3$ we ...
Mare's user avatar
  • 25.8k
5 votes
0 answers
343 views

Interpolated simple integral fusion categories of Lie type

$\DeclareMathOperator\PSL{PSL} \DeclareMathOperator\Rep{Rep}$The idea motivating this post is that there should exist a global understanding of the unitary fusion categories $\Rep(G(q))$, with $G(q)$ ...
Sebastien Palcoux's user avatar
5 votes
0 answers
127 views

On a reference for computing global spectrum of $A_n$-curve singularities, by H.Dao and E.Faber

This question is about chasing down a reference in a paper relating to non-commutative crepant resolutions and Cohen-Macaulay representation theory. Allow me to first give a minor introduction. Let $(...
user160167's user avatar
5 votes
0 answers
153 views

Higher analogue of the Auslander-Bridger transpose

Let $A$ be an Artin algebra and $M$ a module with $Ext^i(M,A)=0$ for $i=1,...,n-2$. Then in case $P_{n-1} \rightarrow ... \rightarrow P_0 \rightarrow M \rightarrow 0$ is the beginning of a minimal ...
Mare's user avatar
  • 25.8k
5 votes
0 answers
245 views

Character tables of finite groups and isomorphism

I'd like to ask the following question: Let $G$ and $H$ be finite groups. Is there a useful criterion involving the ordinary character table which assures that $G$ and $H$ are isomorphic as groups?...
Bernhard Boehmler's user avatar
5 votes
0 answers
130 views

Explicit branching rules from $G(n+m)$ to $G(n) \times G(m)$ (where $G = \operatorname{SL}$, $\operatorname{SO}$ or $\operatorname{Sp}$)

Is there in the literature any explicit combinatorial description of the branching rules from $\operatorname{SL}(n+m)$ to $\operatorname{SL}(n) \times \operatorname{SL}(m)$, from $\operatorname{SO}(n+...
Ilia Smilga's user avatar
  • 1,364
5 votes
0 answers
209 views

Understanding Frobenius's Theorem

Let $\Gamma_g$ be the fundamental group of a closed Riemann surface of genus $g$. Let $G$ be a finite group. Then a theorem of Frobenius states that $$ |\mathrm{Hom}(\Gamma_g, G)|/|G|= \sum_{\chi} \...
Dr. Evil's user avatar
  • 2,641
5 votes
0 answers
98 views

submodules of the exterior algebra

Let $A_{n,q}$ be the exterior algebra of a vector space of dimension $n$ over the finite field $F_q$. Let $a_{n,q}$ be the number of submodules of $A_{n,q}$ (meaning submodules of the $A_{n,q}$-...
Mare's user avatar
  • 25.8k
5 votes
0 answers
113 views

On algebras where all indecomposables have no selfextensions

Let $A$ be a finite dimensional algebra (we can assume it is a connected quiver algebra). Call $A$ extfree in case for every indecomposable $A$-module $M$ we have $\operatorname{Ext}_A^i(M,M)=0$ for ...
Mare's user avatar
  • 25.8k
5 votes
0 answers
187 views

Globalizable Galois representations

Let $\rho$ be a $p$-adic representation of $G=\text{Gal}(\bar{\mathbb{Q}_p}/\mathbb{Q}_p)$. When does $\rho$ extend to a representation of the global galois group? What can be said about the locus ...
safety stegosaurus's user avatar
5 votes
0 answers
229 views

Narayana numbers as character values?

The Catalan numbers show up as character values of the symmetric group: Let $\lambda = (n,n)$, a partition with two parts. Then $\chi^\lambda(1^{2n}) = \frac{1}{n+1}\binom{2n}{n}$, the $n$:th Catalan ...
Per Alexandersson's user avatar
5 votes
0 answers
160 views

The fundamental loopoid?

Let $X$ be a homotopy type (modeled as either a topological space or a simplicial set). We can construct a category as follows: The objects are maps $f,g : S^1 \to X$. A morphism $f \to g$ is a map $S^...
Daniel Barter's user avatar
5 votes
0 answers
120 views

Field of definition of compatible system of Galois representations

Let $K,F$ be number fields and suppose that there is a compatible system of Galois representations $$(\rho_{\lambda} : \text{Gal}(\overline{K}/K) \longrightarrow \text{GL}_n(\overline{F}_\lambda) )$$ ...
Sun Ra's user avatar
  • 173
5 votes
0 answers
370 views

Gelfand pairs in $SO(p,q)$

I am considering the groups $SO(p,q)$ over the reals. And inside it some parabolic subgroup, $P$. It can be the minimal parabolic but that is not the issue. It is well known that $P$ has a Levi ...
Gal Yehoshua's user avatar
5 votes
0 answers
112 views

Extreme no loop conjecture for group algebras

Let $A=KG$ be a group algebra for a finite group $G$. Let $S$ be a simple $A$-module. The extreme no loop conjecture predicts that $Ext_A^1(S,S) \neq 0$ implies $Ext_A^i(S,S) \neq 0$ for infinitely ...
Mare's user avatar
  • 25.8k
5 votes
0 answers
147 views

Weyl Group Action on Littelmann Paths

In his paper "Paths and Root Operators in Representation Theory," Littelmann gives an action of the Weyl group on the set of integral paths via $$ \tilde{s}_\alpha(\pi):= \begin{cases} f^n_\alpha(\pi)...
SamJeralds's user avatar
5 votes
0 answers
140 views

Open problems about Morita and derived invariants

Are there properties of rings of which one does not know whether they are Morita or derived invariances? For a recent such example for Morita invariance, see https://www.sciencedirect.com/science/...
Mare's user avatar
  • 25.8k
5 votes
0 answers
236 views

Drinfeld Polynomial for Yangian $Y(\mathfrak{sl}_2)$

I am looking for a direct proof that a highest weight representation of $Y(\mathfrak{sl}_2)$ is finite-dimensional if its highest weight is determined by a Drinfeld polynomial. The results was ...
Zhihua Chang's user avatar
5 votes
0 answers
90 views

Bound on the sum of projective and injective dimension

Recall that a finite dimensional algebra is called piecewise hereditary in case it is derived equivalent to an abelian hereditary category. In proposition 1.2. of https://link.springer.com/article/10....
Mare's user avatar
  • 25.8k
5 votes
0 answers
122 views

Stable equivalence and stable Auslander algebras

Let $A$ be a representation-finite finite dimensional quiver algebra and $M$ the basic direct sum of all indecomposable $A$-modules. Recall that the Auslander algebra of $A$ is $End_A(M)$ and the ...
Mare's user avatar
  • 25.8k
5 votes
0 answers
51 views

Metric structures making the cohomology into a module over a Lie algebra

The cohomology of a closed Kaehler manifold is an $\mathfrak{sl}_2$-module. I think Verbitsky has shown that the cohomology of a closed hyperkaehler manifold is an $\mathfrak{so}_5$-module. For what ...
geometer's user avatar
5 votes
0 answers
194 views

Lusztig's completion for universal enveloping algebra

In Arkhipov, Bezrukavnikov and Ginzburg's paper "Quantum Groups, the loop Grassmannian and the Springer resolution", they mentioned that Lusztig introduced a certain completion for universal ...
userabc's user avatar
  • 667
5 votes
0 answers
184 views

Homeomorphisms of Springer fibers

Let $V$ be a complex $n$-dimensional vector space and denote by ${\cal F}$ its space of complete flags. Let $g \in Gl(V)$ be unipotent and consider the Springer fiber ${\cal F}_g$ of its fixed points ...
Lucas Seco's user avatar
  • 1,103
5 votes
0 answers
149 views

Group with Character Degrees {1,pq,pr,qr}, where p,q and r are distinct primes

I am currently trying to bound the derived length of certain solvable groups assuming that they have only two irreducible monomial complex character degrees. Using induction, it often suffices to ...
Joakim Færgeman's user avatar
5 votes
0 answers
210 views

Motivation and Difference of Category O Definition for Kac-Moody Algebras

My first encounter with Category $\mathcal{O}$ was (perhaps unusually) learning about Kac-Moody algebras from Kac's book. Kac takes the following definition: The Category $\mathcal{O}$ has objects $\...
SamJeralds's user avatar
5 votes
0 answers
131 views

$q$-Kostant partition function and flow polytopes?

The Kostant partition function is known to be related to volumes and Ehrhart polynomials of flow polytopes of graphs (see e.g. https://link.springer.com/article/10.1007/s00031-008-9019-8 or https://...
Sam Hopkins's user avatar
  • 22.7k
5 votes
0 answers
357 views

$\text{Determinant}=(\sum \text{Determinant})^2$

Denote by $\delta_{n-1}=(n-1,n-2,\dots,1,0,0,\dots)$ the staircase partition and the embedded partition $\lambda=(\lambda_1,\lambda_2,\dots)\subset\delta_{n-1}$. QUESTION 1. Is this true? $$\det\...
T. Amdeberhan's user avatar
5 votes
0 answers
129 views

Identity for classes of plane partitions

There are several classes of plane partitions in the literature. Among these, let's look at the enumeration of three of them: the symmetric (SPP), totally symmetric (TSPP) and totally symmetric and ...
T. Amdeberhan's user avatar
5 votes
0 answers
1k views

Frobenius formula

I know two formulas by the name of Frobenius. The first one computes the number $$\mathcal{N}(G;C_1,\dotsc,C_k):=|\{(c_1,\dotsc,c_k)\in C_1 \times \cdots \times C_k\:|\:c_1\cdots c_k=1\}|,$$ where $...
Gabriel's user avatar
  • 943
5 votes
0 answers
87 views

Existence of anti-symmetric hochschild homology representatives

Let $A$ be an associative algebra over a field $k$. Let $A_{L}$ be the Lie algebra of $A$ with commutator bracket. Then if $M$ is a bimodule for $A$ there is an associated representation of $A_{L}$ ...
user avatar
5 votes
0 answers
156 views

Spinor representation for $\operatorname{Spin}(V \oplus V^*)$

I'm studding Hitchin's Generalized Calabi-Yau Manifolds https://arxiv.org/abs/math/0209099 and I've stuck here: Suppose that $V$ is a vector space and denote its dual by $V^*$. Now we know that the $\...
Parisa Mahmoudi's user avatar
5 votes
0 answers
103 views

Derived invariant acyclic algebras

Call a connected quiver algebra $A=KQ/I$ (finite quiver Q and admissible ideal I) derived-invariant in case every quiver algebra derived equivalent to $A$ is even isomorphic to $A$. For example local ...
Mare's user avatar
  • 25.8k
5 votes
0 answers
129 views

Differential operators on $G/K$

Let $G$ be a connected Lie group and $K$ a compact subgroug of $G$. The question is about the algebra of the differential operators $Diff(G/K)$ on $G/K.$ Let $U(\mathfrak g)$ denote the universal ...
jorge vargas's user avatar
5 votes
0 answers
83 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 ...
Mare's user avatar
  • 25.8k
5 votes
0 answers
206 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 ...
Aegon's user avatar
  • 173
5 votes
0 answers
231 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 ...
Rosa Ivanovic's user avatar
5 votes
0 answers
154 views

lifting of idempotents in group ring

Let $G$ be a finite group, and let $\pi:G\to Q$ be a surjective group homomophism. The map $\pi:G\to Q$ does not necessarily split, but we can always find a set theoretical splitting $s:Q\to G$. In ...
Ehud Meir's user avatar
  • 4,969
5 votes
0 answers
392 views

Mathematical proof of Regge symmetry

In the representation theory of the group $SU_2$ a big role is played by so-called $6j-$symbols. Let me sketch its definition (some other interpretations could be found here). Denote a representation ...
Daniil Rudenko's user avatar

1
57 58
59
60 61
137