All Questions
Tagged with reference-request rt.representation-theory
823 questions
8
votes
2
answers
576
views
Two statistics on the permutation group
Let $\mathfrak{S}_n$ be the permutation group on an $n$-element set. For each fixed $k\in\mathbb{N}$, consider the two sets
$$A_n(k)=\{\sigma\in\mathfrak{S}_n\vert\,\, \text{$\exists i,\,\, 1\leq i\...
- 43.2k
0
votes
0
answers
197
views
'Adelic torus' not arising from a rational torus
Let $G$ be a reductive group over a global field $F$, and $\gamma$ a strongly regular semi-simple element of $G(F)$. Then the centralizer $G_\gamma$ is defines an $F$-torus $T$, and hence by base ...
- 3,799
3
votes
2
answers
1k
views
Rational Characters of a reductive group have the same rank as split component
Let $G$ be a connected reductive group defined over a perfect field $F$. The split component $A$ of $G$ is the unique maximal $F$-split subtorus of the radical of $G$. For an algebraic group $H$ ...
- 6,180
5
votes
1
answer
362
views
Reference for Mackey functors with group value inverted
I'm looking for a reference for the decomposition of the category of Mackey functors for a finite group when the order of the group is inverted. (There is also an analogous decomposition for the ...
- 25.6k
4
votes
1
answer
193
views
Decomposition of $\Gamma$-modules into simple objects
Let $\Gamma$ be the category of finite pointed sets. The abelian category $\mathrm{Mod-}\Gamma$ is the category of functors $\Gamma^{\mathrm{op}} \to \mathrm{Vect}_k$, where $k$ is a field (see ...
- 5,482
1
vote
0
answers
109
views
Reference Request: (Borel) Iwahori Spherical Representations
I was told that Borel had a result about Iwahori Spherical automorphic representations being upper-triangular (/semistable). Where can I find this?
- 1,629
7
votes
0
answers
543
views
Representation theory of symmetric group for dummies
I have to grade a master project on representations of symmetric groups (char $0$) third time in my life and finally I came to a conclusion that I have to get a grasp of the matter. I am aware of ...
- 14.3k
5
votes
0
answers
215
views
Explicit description of the Plancherel measure for $GL_n(\mathbb{R})$
Let $G:=\mathrm{GL}_n(\mathbb{R})$ and $f\in C_c^\infty(G)$. One can uniquely determine the Plancherel measure $d\mu_p$ on $\hat{G}$, the unitary (actually tempered) dual of $G$, by the equation
$$f(g)...
- 1,692
3
votes
0
answers
61
views
Reference request: table of representation rings and relations
Where can one find a table of generator–relator expressions for representation rings $R(G)$ of simple Lie groups $G$ and explicit maps between them? For example, given a maximal torus $T < G$ or a ...
- 2,995
5
votes
1
answer
323
views
Is this result on tensor products of faithful modules known?
On the first page of this preprint, I give a proof (a simplification
of an argument suggested by Ken Goodearl) that if k is a field,
A and B are k-algebras (not necessarily commutative), M is a ...
- 595
6
votes
1
answer
399
views
Exceptional isomorphism with Spin(6,2)?
There are all sorts of curios in low-dimensional Lie groups and Lie algebras, many of them due to the presence of the quaternions. There is, I have recently learned, an isomorphism $SO(6,2) \simeq SO(...
- 35.5k
5
votes
1
answer
394
views
Classification of indecomposable modules in tame hereditary algebras
An algebra $A$ is said to be tame if the isomorphism classes of indecomposable $A$-modules in each dimension occur in a finite number of 1-parameter families. $A$ is said to be of finite ...
- 482
1
vote
0
answers
68
views
Reference for $q^{\eta\sum_{i}t_{\lambda_{i}}\otimes t_{\mu_{i}}}$ part of an expression for universal $R$-matrix of a quantum group algebra
The Wikipedia entry for quantum groups states that a quantum group $U_{q}\left(\mathfrak{g}\right)$ has an infinite formal sum that plays the role of an $R$ matrix, which is the product of two factors,...
- 357
17
votes
1
answer
502
views
For which finite groups $G$ is every character a virtual permutation character?
Let $G$ be a finite group. A (complex) character $\chi$ of $G$ is said to be a virtual permutation character if it can be expressed as a $\mathbb{Z}$-linear combination of characters induced from the ...
- 1,661
8
votes
3
answers
419
views
For which finite groups $G$ does the Wedderburn decomposition of $\mathbb{Q}[G]$ consist only of fields and division algebras?
Let $G$ be a finite group. Then the rational group algebra $\mathbb{Q}[G]$ has a wedderburn decomposition of the form $\prod_i M_{n_i}(D_i)$ where each $D_i$ is a division algebra.
My question is: ...
- 1,661
11
votes
1
answer
480
views
Uncle of Witt algebra
A Witt algebra W is an infinite-dimensional Lie-algebra defined by the generator relations:
W: $[L_{j},L_{k}]:=(j-k)\cdot L_{j+k}$
And my first thought was: What about the analogous algebra defined by
...
- 4,829
10
votes
1
answer
802
views
Restriction of irreducible unitary representation to normal subgroup of finite index
Let $G$ be a Lie group (or more generally a locally compact group), let $N$ be a closed and normal subgroup of $G$ of finite index. Let $H$ be an infinite dimensional complex Hilbert space, and let $\...
- 333
4
votes
1
answer
783
views
Three dimensional representations of Alternating group
The alternating group $A_5$ has $2$ irreducible representation of degree $3$. The characters for these representations have irrational values. I guess the ring of invariants of these representations ...
- 125
7
votes
2
answers
704
views
Reference for nonlinearity of covers of $\operatorname{SL}(2,\mathbb R)$
It is known that no nontrivial connected cover of $\operatorname{SL}(2,\mathbb R)$ admits a faithful finite dimensional linear representation (see, for example, page 143 in Fulton-Harris and Exercise ...
- 1,782
11
votes
3
answers
2k
views
Text for studying group representations in the context of (abstract) harmonic analysis
I would like to study elements of representation theory as I often encounter it when reading texts on harmonic analysis. I was therefore curious if someone could recommend a book for this.
When ...
user92170
3
votes
0
answers
188
views
Bott-type vanishing results for the weighted Grassmannian wGr(2,5)
If $G=Gr(k,n)$ denotes the Grassmannian of k-dimensional subspaces in $V= \mathbb C^n$, representation theory gives us a Bott-type result for the cohomology groups $H^q(G, \Omega^p(k))$ of the twisted ...
- 776
4
votes
0
answers
429
views
Auslander-Reiten-Quivers of representation-finite algebras having different 3-dimensional forms
I am looking for references, where I can find (pictures of) connected Auslander-Reiten-Quivers of representation-finite $k$-algebras ($k$ is a (preferably, but not necessarily finite) field) with one ...
- 1,755
3
votes
1
answer
195
views
How to embed $S^2\mathbb{C}^2$ into $S^2S^3\mathbb{C}^2$ and get the ideal of the twisted cubic?
Let $X:=x^3$, $Y:=x^2y$, $Z:=xy^2$ and $W:=y^3$ be the 4 independent generators of $S^3\mathbb{C}^2$, and observe that the kernel of the natural epimorphism (total symmetrisation)
$$
p:S^2S^3\mathbb{C}...
- 2,527
4
votes
0
answers
318
views
Action of orthogonal group on the free Lie algebra
This question is somewhat related and inspired by this post of professor Montgomery.
The free Lie algebra $L(V)$ generated by an $r$-dimensional vector space $V$ is, in the language of https://en....
- 277
6
votes
1
answer
227
views
Quantum group representations from (convolution) matrix units?
Let $A=F(\mathbb{G})$ be the algebra of functions on a finite quantum group with a Haar state $$h=:\int_\mathbb{G}:F(\mathbb{G})\rightarrow \mathbb{C}.$$
There is a convolution product on $A=F(\...
- 1,027
5
votes
1
answer
525
views
what is the injective hull of indecomposable module of preprojective algebra
Let $Q$ be a ADE type quiver and $s_i$ ($i$ runs through the vertices of $Q$) be the simple $\Lambda$-module with 1-dimensional vector space at vertex $i$ and zero-dim at other vertices. Here $\Lambda$...
- 849
2
votes
0
answers
414
views
Mixed up by definitions of mildly mixing
Here are two setup where the notion of "mildly mixing" comes up: for representations and for group acting by measure preserving transformations (see definitions below). Since a natural class of ...
- 4,432
1
vote
1
answer
172
views
Why are exchange graphs of quivers with the same underlying graph but have different orientations isomorphic?
I know the fact that (undirected) exchange graphs of quivers with the same underlying undirected graph but have different orientations are isomorphic (i.e. quivers that are just finitely many arrow-...
- 417
1
vote
1
answer
220
views
Reference request: $\chi^{\lambda'}(\sigma) = (-1)^{n-\ell(\sigma)} \chi^\lambda(\sigma),$ for characters of the symmetric group
I'm looking for a text I could cite that explicitly states the following result: for $\chi^\lambda$ the irreducible character of the symmetric group indexed by the partition $\lambda$, and for $\sigma ...
- 2,151
5
votes
1
answer
299
views
Sum of skew characters over hooks and "odd" partitions
Let us call a partition odd if all its parts are odd, and let $Odd(n)$ be the set of all odd partitions of $n$, e.g. $Odd(6)=\{(5\,1),(3\, 3),(3\,1^3),(1^6)\}$.
Let $H(n)$ denote the set of all hook ...
- 2,552
2
votes
1
answer
251
views
Characterization of restricted weights of representations of real semisimple Lie groups
I need to use the following theorem:
Let $\mathfrak{g}$ be a semisimple real Lie algebra, $\Sigma$ a set of restricted roots for $\mathfrak{g}$. Let $\rho$ be any finite-dimensional representation of ...
- 1,574
3
votes
2
answers
529
views
What does the regular representation of the coinvariant ring of a unitary reflection group look like?
Let $V$ be a complex vector space of finite dimension $n$ and let $W$ be a finite unitary reflection group. This is, $W$ is a subgroup of $GL(V)$ generated by reflections, i.e., elements $r \in GL(V)$ ...
- 3,271
0
votes
1
answer
290
views
Whitehead's lemma (Lie algebras) for reductive Lie algebras [closed]
Whitehead's lemma (Lie algebras) is: Let $\mathfrak{g}$ be a semisimple Lie algebra over a field of characteristic zero, $V$ a finite-dimensional module over it and $f$: $\mathfrak{g} \to V$ a linear ...
- 6,201
4
votes
1
answer
2k
views
Reference request for an introduction to deformation theory in algebraic geometry
I'd like some introductory references for deformation theory in algebraic geometry. I'm interested in survey articles too but I primarily want references which give all the definitions and go through ...
6
votes
0
answers
236
views
Subquotients of Jantzen Filtration for Kac-Moody algebras
Let $\mathfrak{g}$ be a complex symmetrizable Kac-Moody algebra, with triangular decomposition $\mathfrak{n}^- \oplus \mathfrak{h} \oplus \mathfrak{n}^+$. Let $\lambda \in \mathfrak{h}^*$, and $M(\...
- 329
4
votes
1
answer
215
views
Does a $G$-structure on $M$ automatically descend to a contact $G$-structure on $\mathbb{P}T^*M$?
If $M$ is a real smooth manifold of dimension $n+1$, by $D\in\mathbb{P}T^*M$ I mean a tangent hyperplane at some point of $M$. I denote by $b$ the canonical projection of the $(2n+1)$-dimensional ...
- 2,527
4
votes
0
answers
367
views
Representation theory and associated bundles
I am looking for a text or set of notes that discusses the relationship between the representation theory of Lie groups and associated vector bundles, preferably using modern categorical language. For ...
- 6,025
6
votes
0
answers
261
views
Local character expansion for discrete series representations of $GL_n(F)$
I'm interested about what, if anything, is known about the local character expansion of discrete-series representations of $GL_n(F)$, where $F$ is a $p$-adic field.
First, some notation: let $G$ be a ...
- 1,453
6
votes
0
answers
416
views
Bott-Samelson construction of a perfect Morse function on G/T
An undergraduate student of mine is interested in writing a "senior" thesis on the topology of Lie groups. Let $G$ be a simply connected compact Lie group and $T$ a maximal torus. One way of ...
- 1,331
10
votes
3
answers
508
views
Construct discrete series of SL(2,R) as kernel of twisted Dirac operators
I’m studying the paper of (Baum-Connes-Higson, ex 4.25), and I would like to give an explicit computation for the Connes-Kasparov conjecture for SL(2,R).
The idea is that each non-trivial ...
- 288
4
votes
1
answer
421
views
Birch's conjecture from Representation Theory
Birch has a conjecture about which automorphic forms on $PGL(2)$ are the lifts from nonsplit $O(3)$. Temporarily ignore global issues, and focus on the local nonarchimedian picture. The automorphic ...
- 2,429
4
votes
1
answer
260
views
Reference request: normalization of intertwining operators for GL(2, C)
Take $F$ a local field and $\chi_1, \chi_2$ two characters, write $M(\chi_1, \chi_2)$ for the standard intertwining integral
$$M(\chi_1. \chi_2).f(g) := \int_{F} f\left( \begin{pmatrix} 0&-1\\ 1&...
- 590
4
votes
0
answers
202
views
Shifts in the decomposition of Bott-Samelson bimodules
Let $k$ be an algebraically closed field of characteristic $0$, let $V=k^n$ be a $k$ vector space of dimension $n$, and let $R=k[V]$ be the ring of polynomial functions on $V$. Suppose that $W\subset\...
- 878
8
votes
1
answer
549
views
Ring of invariants for the regular representation
The symmetric group $S_n$ acts on $\mathbb C^n$ by permuting the coordinates. In this case the ring of invariants is generated by elementary symmetric polynomials in n-variables. Now consider the ...
- 195
4
votes
1
answer
273
views
Homological characterisation of standardly stratified algebras using Ext
Let A be a finite dimensional algebra and $S_1,S_2,...,S_n$ the simple $A$-modules and $P_1,..,P_n$ the indecomposable projective $A$-modules. For $i=1,...,n$, define the standard module $\Delta_i$ as ...
- 26.5k
6
votes
0
answers
209
views
Classification of representation-finite algebras up to stable equivalence of Morita type
Assume $K$ is an algebraically closed field.
I wanted to ask if there is a classification of the representation-finite $K$-algebras up to stable equivalence of Morita type (at least for some small ...
- 26.5k
9
votes
2
answers
787
views
Request for classical articles in representation theory
I am planning in running a Ph.D. student seminar next year on representation theory in the spirit of MIT Kan's Seminar where students give lectures on classical articles on representation theory that ...
2
votes
0
answers
360
views
Differential of the adjoint quotient map
My question is regarding a paper by R.W Richardson titled "Derivatives of invariant polynomials on a semisimple Lie Algebra" ** . In this paper, he reports on computations of the rank of the ...
- 1,073
2
votes
1
answer
736
views
reference on representation theory of SO*(2n)
I'm interested in the representation theory of the non-compact real Lie group $\mathrm{SO}^*(2n)$, the subgroup of matrices $M\in\mathrm{SO}(2n,\mathbb{C})$ satisfying
$$
M^\dagger\eta \,M=\eta,\qquad ...
18
votes
3
answers
1k
views
What happened to the fourth paper in the series "On the classification of primitive ideals for complex classical Lie algebras" by Garfinkle?
In a series of papers in Compositio Math. entitled On the classification of primitive ideals for complex classical Lie algebras I, II and III, Garfinkle describes an algorithm that allows one to ...
- 2,508