All Questions
Tagged with reference-request rt.representation-theory
823 questions
4
votes
0
answers
158
views
Relation between two Harish-Chandra homomorphisms
Let $\mathfrak{g}$ be a Lie algebra admitting a triangular decomposition $\frak g = N^-\oplus h\oplus N^+$ and $\gamma$ the classic Harish-Chandra isomorphism defined on the center $\frak Z$ of the ...
- 141
3
votes
0
answers
311
views
What is known about representations of $S_n$ in other categories?
Is anything known about representations of the symmetric group $S_n$ for categories other than $\textbf{Vect}_k$, vector spaces and linear maps over a field $k$.
That is, a group $G$ can be considered ...
- 571
1
vote
2
answers
350
views
Finding lectures PDF "Four lectures on simple groups and singularities"
I would be very interested to find the PDF "Four lectures on simple groups and singularities" by Peter Slodowy, especially the lecture 4. I used to print them but lost it. Does anyone has ...
2
votes
0
answers
102
views
Category O for (Yangian) toroidal Lie algebras?
Suppose throughout that $g$ is a finite-dimensional simple Lie algebra over $\mathbb{C}$ and let us denote:
$$g_{[2]} := g \otimes_{\mathbb{C}} \mathbb{C}[v^{\pm 1}, t^{\pm 1}]$$
$$g_{[2]}^+ := g \...
- 1,516
6
votes
1
answer
1k
views
Full expansion of $\det(I+\varepsilon A)$
It is well known that given a $n \times n$ matrix $A$, it holds that
$$ \det(I + \varepsilon A)= 1 + \varepsilon \operatorname{tr}(A) + O(\varepsilon ^2).$$
I would need a full representation of $ \...
- 697
2
votes
0
answers
137
views
$p$-adic Banach group algebra
Let $G$ be a discrete group. Consider the Banach $\mathbb{Z}_p$-algebra: $$c_0(G, \mathbb{Z}_p) = \{ F : G \to \mathbb{Z}_p \mid \lim_{g \to \infty} |F(g)|_p = 0 \}$$ with the product given by the ...
- 1,485
1
vote
1
answer
214
views
Conditions of P for existence of orthogonal matrix Q and permutation matrix U satisfying QP = PU
Question: Let $P\in \mathbb{R}^{d\times n}$ be a $d$-rank real matrix and $PP^T = c I_d$ with a certain constant $c > 0$. Under what additional conditions of $P$ does there exist an orthogonal ...
- 187
1
vote
0
answers
106
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Uniqueness of indecomposable decomposition (Krull–Schmidt) for finitely generated modules over commutative Noetherian standard graded rings
Let $R_0=\mathbb C$ and $R=\bigoplus_{i\geq 0} R_i$ be a commutative Noetherian graded ring such that the grading is standard, i.e., $R=R_0[R_1]$. Let $M$ be a finitely generated $R$-module. Evidently,...
- 412
5
votes
0
answers
293
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On the deformation theory of associative algebras
Let us start by recalling the notion of a formal deformation:
Let $K$ be a field of characteristic zero and $A$ be an associative $K$-algebra. Consider a commutative augmented $K$-algebra $R$, with ...
- 541
1
vote
0
answers
64
views
How to find all linear transformations commuting with all elements of the image of an isotypic representation of a compact Lie group
Let a linear finite-dimensional isotypic (i.e., decomposable into a direct sum of isomorphic irreducible subrepresentations) representation of a compact Lie group $G$ in a finite-dimensional space $V$ ...
- 309
2
votes
0
answers
88
views
Simple modules and trivial source modules
Let $(K,\mathcal{O},k)$ be a large enough $p$-modular system.
In this context, I would like to ask what is known about the following question:
when are simple $kG$-modules trivial source modules?
So ...
- 1,755
16
votes
1
answer
408
views
Is there a relationship between Broué's abelian defect group conjecture and Alperin's weight conjecture?
Let $G$ be a finite group, let $k$ be a large enough field of characteristic $p>0$. Let $p\mid |G|$.
Broué's abelian defect group conjecture states the following:
Let $B$ be a block of $kG$ with
...
- 1,755
1
vote
0
answers
83
views
$p$-modular splitting systems and the characteristic of the ring $\mathcal{O}$
Let $k=\overline{k}$ be a field of characteristic $p$.
Let $(K,\mathcal{O},k)$ be a $p$-modular system.
Let both $k$ and $K$ be splitting fields for $G$ and its subgroups.
The ring $\mathcal{O}$ can ...
- 311
8
votes
1
answer
356
views
Homological conjectures for finite dimensional commutative algebras
$\DeclareMathOperator\Ext{Ext}\DeclareMathOperator\Hom{Hom}$>Question: What are some (open) homological conjectures that are also relevent for finite dimensional commutative algebras over a field $...
- 26.5k
7
votes
2
answers
376
views
Basis parametrized by the symmetric group elements for the coinvariant algebra
Let $A_n$ be the coinvariant algebra of the symmetric group $S_n$.
This algebra has vector space dimension $n!$.
$A_n$ is the quotient algebra of the polynomial ring $K[x_1,...,x_n]$ by the elementary ...
- 26.5k
7
votes
0
answers
176
views
The quotient of a higher Specht polynomial over the corresponding regular Specht polynomial
I'll need some notation before I can phrase my question, so please bare with me for a little. I'll try to get there as fast as possible (it's also my first MO question...).
Let $\lambda$ be a ...
- 131
5
votes
1
answer
505
views
Generalized Wigner 3-j symbol and Legendre functions
Let $P_{n}(x)$ the $n-th$ Legendre polynomial. It is well-knonw that $$\int_{-1}^1 P_n(x) P_m(x) P_h(x) \, dx=2\left(\begin{array}{ccc}
n & m & h\\
0 & 0 & 0
\end{array}\right)^{2}\tag{...
- 219
9
votes
0
answers
286
views
Is Landvogt's thesis "The functorial properties of the Bruhat–Tits building" available online?
Universität Münster publishes theses online through "miami", but "miami" doesn't have Erasmus Landvogt's thesis (search).
ProQuest (predatorily) provides many theses, but they don'...
- 12.9k
5
votes
1
answer
268
views
Enumerating monomials in Schur polynomials
Let $s_{\lambda}(x_1,\dots,x_k)$ be the Schur polynomial associated to the partition $\lambda=(\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_k>0)$.
Among the many things involved with these ...
- 43.2k
0
votes
0
answers
99
views
General reference for finite dimensional $*$-algebras over $\mathbb R$?
What references are there for studying finite-dimensional $*$-algebras over the field $\mathbb R$ in their full generality? We assume these are associative and unital.
Note that:
Not every algebra ...
- 4,943
6
votes
0
answers
236
views
Group homomorphism from $\mathrm{GL}_p$ to $\mathrm{SL}_p$ in characteristic $p$
If $k$ is a commutative field of characteristic $p>0$, then the map
$$ \theta \colon \mathrm{GL}_p(k) \to \mathrm{SL}_p(k) \colon A = (a_{ij}) \mapsto (\det A)^{-1} (a_{ij}^p) $$
is a group ...
- 6,614
4
votes
2
answers
378
views
Splitting field for $\mathrm{GL}(2,p)$ - reference request
It seems to me from a quick glance at several sources describing the complex and modular irreducible representations of $\mathrm{GL}(2,p)$ that any field $K$ containing a primitive $(p-1)$-root of ...
- 38.6k
2
votes
0
answers
228
views
Ramanujan's theta functions and hook lengths?
Given an integer partition $\lambda\vdash n$ of $n$, one may associate a Young diagram $Y(\lambda)$ to it followed by a computation of hook length $h_{\square}$ for each cell $\square=(i,j)$ in $Y(\...
- 43.2k
6
votes
1
answer
588
views
A numerical matrix of power sum polynomials
Let $p_i=x_1^i+x_2^i+\cdots+x_m^i=\sum_{k=1}^mx_k^i$ be the power sum polynomials. Then, the determinant of the $m\times m$ Hankel matrix $M_m=(p_{i+j-2})$, for $1\leq i,j\leq m$, has a neat ...
- 43.2k
1
vote
0
answers
59
views
K-finiteness of unitary representations of Poincaré-like groups?
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\ISO{ISO}$I'd like to know if there are any papers that study the following problems:
Determine when decomposing the unitary irreps of $\ISO(d,1)$ into ...
- 144
3
votes
1
answer
140
views
Asymptotics of Haar moments on general Lie groups
I am trying to understand the asymptotics of Haar Moments on general compact Lie groups (in particular, subgroups of $\mathrm{SU}(n)$). I have learned that closed form formulae for these moments are ...
- 179
3
votes
0
answers
361
views
References on $p$-adic Langlands
As a student with a background in Algebraic Geometry (up to chapter 3 of Hartshorne) and basic Algebraic Number Theory, where should I begin learning about the $p$-adic Langlands program? What are the ...
- 1,485
2
votes
1
answer
258
views
Literature request: $K^b(\text{proj} A)$ Krull-Schmidt for $\text{gl dim}A = \infty$ and general results about its Grothendieck group
I'm interested in the Grothedieck group of the triangulated category $K^b(\text{proj}A)$ when $A$ is a finite dimensional algebra over a field of infinite global dimension.
For this purpose, It would ...
- 123
5
votes
1
answer
318
views
Up to date summary on semisimple Hopf algebra over $\mathbb{C}$
Question: Is there an up to date summary of results on the classification of semisimple Hopf algebras over $\mathbb{C}$ (or a field of characteristic 0)?
Here are some questions I wonder about:
...
- 26.5k
0
votes
1
answer
349
views
Log associahedra and log noncrossing partitions--raising ops and symmetric function theory for $A_n$ (references)
Where do the following three sets $[LA]$, $[ILA]$, and $[LN]$ of partition polynomials appear in the literature?
There are two sets of partition polynomials, not in the OEIS, that serve as the ...
- 10.5k
2
votes
0
answers
134
views
Algebra of finite width matrices
$\DeclareMathOperator\FWM{FWM}\DeclareMathOperator\End{End}$For any ring $R$ there's an algebra of finite width matrices with entries in $R$. By finite width matrices I mean the ones that have only ...
- 4,600
3
votes
1
answer
173
views
$\Omega$ for noetherian semiperfect rings
Let $A$ be a a two-sided noetherian semiperfect ring and assume that the injective dimension of the left and right regular modules are equal to $n \geq 1$.
Let $\Omega^n(mod A)$ be the category of $n$-...
- 26.5k
4
votes
1
answer
185
views
Frobenius series for the $S_n$-module $\mathbb{Q}[X]$
I'm reposting this question, by recommendation of a moderator.
I'm reading Haiman's article titled Conjectures on the quotient ring by diagonal invariants. In what follows, all vector spaces and ...
- 141
5
votes
0
answers
207
views
parameter of a quantum group
I am currently learning about quantum groups, and I got a question about how two different ways of thinking the $q$-parameter of quantum groups are related to each other. Here by a quantum group, I ...
- 323
4
votes
0
answers
177
views
Reference for Iwahori-Hecke algebras
I recently came across the notion of an Iwahori-Hecke algebra. I would like to learn the basics about this type of algebras (mainly to get an intuition about them, as they seem to be related to some ...
- 541
2
votes
2
answers
680
views
Complete representation theory of $\mathrm{SL}(2,\mathbb R)$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Is the complete representation theory of $\SL(2,\mathbb R)$, $\GL(2,\mathbb R)$, $\SL(2,\mathbb C)$, and $\GL(2,\mathbb C)$ known, in the sense ...
- 3,873
1
vote
1
answer
438
views
Categories associated to digraphs
Let's take a directed graph, or a digraph, $G=(V,E)$ given by a finite set of vertices $V$ and a finite set of edges $E$. We can assume that pairs of vertices can have parallel edges between them and ...
2
votes
1
answer
214
views
A Vandermonde like determinant with exponentials
Let $n\geq m$ be non negative integers, and consider a list of $(n+m+1)$ distinct numbers (complex or real). I am interested in getting a closed form formula for the following determinant: $\det\left[\...
- 275
3
votes
1
answer
102
views
References on coefficient quivers
I would like to study about coefficient quivers, but I cannot find a good reference, as book for example. I could find many papers working with coefficient quivers, but none of them give a book or a &...
- 35
2
votes
0
answers
138
views
Construction of a certain long exact sequence
Let $A$ be a noetherian ring (not necessarily commutative) or for simplicity even a finite dimensional algebra over a field.
Let $X$ and $U$ be finitely generated $A$-modules and let $add(U)$ be the ...
- 26.5k
5
votes
1
answer
207
views
Iwahori-spherical tempered representations with fixed vectors for larger parahoric subgroups
Let $F$ be a $p$-adic field and $G$ be split over $F$ plus all other usual adjectives. Let $\pi$ be an admissible tempered representation with Iwahori-fixed vectors. Let $P\supset I$ be some parahoric ...
- 413
10
votes
1
answer
399
views
Basic algebra of $\mathcal{O}_0(\mathfrak{sl}_n(\mathbb{C}))$ — Reference request
It is well known that the principal block $\mathcal{O}_0$ of the BGG category $\mathcal{O}$ of a semisimple Lie algebra is equivalent to the category of finitely generated modules over a certain ...
- 103
3
votes
0
answers
463
views
Representations of triangle groups
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\PSL{PSL}$I am self-studying triangle groups and the following question comes up.
Let $G$ denotes $(2,3,7)$ triangle group. It is symmetry group of $(2,...
- 613
13
votes
1
answer
853
views
Applications of equivariant homotopy theory to representation theory
Equivariant homotopy theory focuses on spaces together with some group action on them. Jeroen van der Meer and Richard Wong have a paper where they use equivariant methods to compute the Picard group ...
- 404
2
votes
0
answers
98
views
Question concerning relationships between different $p$-modular systems and Brauer character values
Let $(K,\mathcal{O},k)$ be a large enough $p$-modular system, where $\mathcal{O}$ is a complete discrete valuation ring of characteristic zero with unique maximal ideal $J(\mathcal{O})$, algebraically ...
- 1,755
4
votes
1
answer
273
views
Wedderburn decomposition of special linear groups
$\DeclareMathOperator\SL{SL}\newcommand\card[1]{\lvert#1\rvert}$I want to study about Wedderburn decomposition of group algebra $k\SL(n,\mathbb{F}_p)$ where $k$ is either an algebraically closed field ...
- 249
18
votes
2
answers
1k
views
Has anyone seen this construction of the Weil representation of $\mathrm{Sp}_{2k}(\mathbb{F}_p)$?
$\def\FF{\mathbb{F}}\def\CC{\mathbb{C}}\def\QQ{\mathbb{Q}}\def\Sp{\text{Sp}}\def\SL{\text{SL}}\def\GL{\text{GL}}\def\PGL{\text{PGL}}$Let $p$ be an odd prime. The Weil representation is a $p^k$-...
- 156k
7
votes
1
answer
185
views
Origin of the abacus bijection
What is the origin of the abacus bijection (aka the rim hook bijection, aka the Stanton-White bijection, aka James's bijection)?
Igor Pak, in his 2000 article "Ribbon tile invariants" (...
- 19.7k
3
votes
1
answer
106
views
Reference request for equivalent formulations of being absolutely indecomposable
I would like to ask the following question.
I am searching for a reference for the following statement:
Suppose $k$ is a perfect field. Let $A$ be a (symmetric) $k$-algebra and let $M$ be a finitely ...
- 311
5
votes
1
answer
61
views
Do $F$-traces of simple modules at $p'$-classes uniquely determine the module?
Let $G$ be a finite group and let $p$ be a prime number such that $p\mid |G|$.
Let $\text{IBr}(G)$ denote the set of irreducible Brauer characters of $G$ for the prime $p$.
Assume $\mathbb{F}_{q}$ is ...
- 311