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

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3
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1answer
150 views

Do the following two filtrations of the affine Grassmannian agree?

Let $H = L^{2}(S^{1},\mathbb{C}^{n})$, $H_{0}\subseteq H$ the subset of maps that extend holomorphically to the unit disc, and $H_{m} = z^{m}H_{0}$. Consider the affine Grassmannian for $GL_{n}$ in ...
1
vote
0answers
69 views

research on the structure/properties of permutation matrix/table with $(i,j)th$ entry as $\pi_j\circ \pi_i^{-1}$

Is there any research on the structure/properties of permutation matrix/table with $(i,j)th$ entry as $\pi_j\circ \pi_i^{-1}$, where $\{\pi_1,\pi_2,...,\pi_{k!}\}=S_k$? I know if we apply the ...
6
votes
2answers
293 views

Global Affine Flag Variety and Affine Flag Variety

There is a construction of a global affine flag variety over $\mathbb{A}^1$ (or another curve) $Fl_{\mathbb{A}_1}$ such that each fiber above $\epsilon \neq 0$ is isomorphic to a direct product of the ...
4
votes
1answer
350 views

Tannakian fundamental group of two explicit tensor categories

Let $K/k$ is a field extension and $G$ an affine group scheme over $K$. What are the Tannakian fundamental groups of these two $k$-tensor categories (with trivial fiber functors over $k$): 1. The ...
0
votes
1answer
101 views

Graph lifts and representation theory

Is there any connection known between the two? One can naturally define lifts of graphs by groups like $\mathbb{Z}_k$ and hence I wonder if representation theoretic properties can be used to say ...
2
votes
0answers
63 views

Mellin transform of Plancherel measure

Let $G$ be a reductive p-adic group with a chosen Haar measure $dg$. The Plancherel measure is the measure $\mu$ on the set of (tempered) irreducible representations of $G$ such that for any locally ...
3
votes
0answers
106 views

Invariant Laurent polynomials under cyclic group action

Start with the cyclic group $G:=\mathbb{Z}/p$ of prime order $p$ and and an integer lattice $P:=\mathbb{Z}^p$. Let $G$ act on $P$ by cyclic permutation of coordinates. There is an induced action on ...
17
votes
2answers
819 views

What is modular representation theory for groups good for?

I am an absolute beginner in modular representation theory of finite groups. I know some things in representation theory in characteristic zero. My questions are regarding the main goals of this part ...
3
votes
1answer
219 views

Explicit Isomorphism between $Cl(8)$ and $\mathbb{R}(16)$

I am looking for a explicit isomorphism between $Cl(8)$ (Clifford algebra over $\mathbb{R}^8$ with standard Euclidean metric) and $\mathbb{R}(16)$ (algebra of $16\times 16$ matrices over ...
5
votes
1answer
459 views

exceptional cases in Kazhdan-Lusztig

The Kazhdan-Lusztig story doesn't apply to the four exceptional cases $(E_6)_1$, $(E_7)_1$, $(E_8)_1$, $(E_8)_2$ (see this earlier question of mine). What's special about those cases?
2
votes
0answers
85 views

Is specht module the intersection of two induced modules?

I heard someone said( maybe Okonov) that specht module is the intersection of two induced modules, but I do not know why.The details of my question is as follows. Let $\lambda\vdash n$ be a partiton, ...
4
votes
0answers
138 views

Number of Irreducible Representations of $U_q(n)$ of Dimension $n$?

For quantum group $U_q(n)$, is it true that it has exactly two non-isomorphic irreducible corepresentations with dimension $n$, and that one is the dual of the other? I know result is in the Chapter ...
1
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0answers
128 views

representation of $SO(p)\times SO(q)$ with $p,q$ odd

Assume $p,q$ odd. We denote by $\sigma_p$ the standard representation of $SO(p)$, that is the representation of $SO(p)$ acting on $\mathbf{R}^p$ as matrix. So is $\sigma_q$. Take $K=SO(p)\times ...
5
votes
2answers
223 views

Embedding $G$ in a $Z(G)$ extension of $\operatorname{Aut}G$

This question follows up a question I asked on math.SE. This is a refinement and a reference request. For what groups $G$ does there exist a $Z(G)$-extension of $\operatorname{Aut}G$ (call it ...
5
votes
1answer
202 views

Intersections of $B$ and $B^-$ orbits in the flag variety $G/B$

Let $G = SL_n(\mathbb{C})$, $B$ be a Borel subgroup, and $B^-$ be the opposite Borel. Both the $B$ and $B^-$ orbits on the flag variety $G/B$ are indexed by the Weyl group $W$. Let $S_{w_1}$ and ...
1
vote
2answers
85 views

Non-degenerate characters of the unitriangular group $U$

I made a previous post which was unclear and mistaken in fundamental aspects, so that it was actually more worthy making this new post than actually editing the previous one. I'm studying the ...
2
votes
1answer
202 views

highest weight representations inside tensor product

Let $G$ be a semisimple simply connected group over an algebraically closed field $k$ of characteristic zero, $B$ a Borel and $T$ a maximal torus. Let $\lambda,\mu,\nu$ be dominant characters of $T$. ...
4
votes
1answer
159 views

Is the restriction of a representation semisimple?

Let $F$ be local field of characteristic zero and $\pi$ be a irreducible admissible representation of $GL_n(F)$. Let us consider its restriction to $GL_{n-1}(F)$. Then I want to know whether ...
1
vote
0answers
88 views

Compatibility of two definitions of Koszul dual

Let $k$ be a field and $A$ a nonnegatively graded ring over $k$. Assume $A_0 = k.$ We have a bigrading on $\operatorname{Ext}(k,k)$ (one corresponding to homological degree, one corresponding to the ...
9
votes
1answer
335 views

Gabriel's theorem over a commutative ring

Is Gabriel's theorem on the indecomposables of representations of quivers of finite type true over a commutative ring, i.e. not necessarily a field?
0
votes
0answers
59 views

adjoint quotient and points in DVRs

Let $G$ be a connected reductive group over an algebraically closed field $k$, $T$ a maximal torus and $W$ its Weyl group. We have a Steinberg map $\chi:G\rightarrow \mathfrak{C}:=T/W$ if we have a ...
1
vote
2answers
110 views

irreducibility of certain subspaces of the permutation group in quantum mechanics

Let $P_j$, $j = 1, \dotsc, N!$ be a set of unitary operators constituting a representation of the symmetric group $S_N$, acting in a sub-Hilbert space $V_0 \subseteq H$ (of a separable Hilbert space ...
1
vote
0answers
121 views

Young symmetrizers question

Let $\lambda$ be a partition of $n$, and let $T$ be the standard tableau associated to $\lambda$ (write the Young diagram of $\lambda$ down and fill in the boxes with $1$ through $n$ left to right, ...
2
votes
1answer
93 views

Rational Points of a Quotient of a Reductive Group by a Parabolic Subgroup

Let $G$ be a reductive group and let $P$ be a parabolic subgroup of $G$ all defined over $\mathbb{Z}$. Also, let $F$ be a number field, is it true (and if so, please provide a reference) that $$ ...
3
votes
1answer
176 views

Are norm-continuous representations smooth?

Let $G$ be a real Lie group and $A$ a unital Banach algebra. Let us call a map $\varphi:G\to A$ a (norm-)continuous representation, if it is continuous $$ x_i\to x\quad\Longrightarrow\quad ...
3
votes
1answer
169 views

How to think about the simple reflection s_0 in the affine Weyl group?

Let $G$ be a simply connected algebraic group over $\mathbb{C}$, $W$ be the Weyl group for $G$ and $W_{aff}$ be the affine Weyl group for the loop group $G(\mathbb{C}((t)))$, $\Phi$ be the coweight ...
6
votes
1answer
284 views

Are the distributive permutation groups linearly primitive?

An action of a group $G$ on a set $X \neq \emptyset$ is called transitive if $\forall x,y \in X$, $\exists g \in G$ such that $g.x = y$. It is called primitive if it is transitive and preserves no ...
8
votes
0answers
120 views

Intersection of Springer fibre and Schubert cell

Let us consider intersections of Springer fibres and Schubert cells in type A. Let $ Y : \mathbb C^n \rightarrow \mathbb C^n $ be a nilpotent operator. Let $$ F_Y = \{ V_0 = 0 \subset V_1 \subset ...
0
votes
1answer
130 views

The coproducts $\mathbb{C}_q[U] \to \mathbb{C}_q[U] \otimes \mathbb{C}_q[U]$ and $\mathbb{C}[U] \to \mathbb{C}[U] \otimes \mathbb{C}[U]$

A coproduct $\varphi: \mathbb{C}_q[U] \to \mathbb{C}_q[U] \otimes \mathbb{C}_q[U]$ is given by: $x \mapsto 1 \times x + x \otimes 1$, where $x$ is a generator of $\mathbb{C}_q[U]$. There is a ...
0
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0answers
40 views

References about reality of minimal affinizations of quantum affine algebras

Let $U_q(\widehat{\mathfrak{g}})$ be the quantum affine algebra associated to a complex simple Lie algebra $\mathfrak{g}$. A simple module $M$ of $U_q(\widehat{\mathfrak{g}})$ is called real if $M ...
4
votes
0answers
104 views

Papers/Programs for computing periodic KL polynomials?

Periodic Kazhdan-Lusztig polynomials (for an affine Weyl group) are polynomials that control Jordan-Holder multiplicities for certain representations ("baby Verma modules") of an algebraic group in ...
0
votes
0answers
104 views

Connection between Lie algebras and fusion rings

Example: Take the irreps of $SU(2)$: $0,1/2,1,...$ (Spin notation.) The quantum dimensions are $1,q+1/q,q^2+q+1+1/q+1/q^2,...$. At $q=(-1)^{1/5}$ this evaluates to $1,\phi,0,...$ and you get the ...
5
votes
1answer
174 views

What is the Schur multiplier of the affine linear group AGL(n,q)?

What is the Schur multiplier of the $n$-dimensional affine linear group $\mathrm{AGL}(n,q)$ over the Galois field with $q$ elements? I am particularly interested in the simple case $n=1$. Computation ...
8
votes
1answer
240 views

Characterization of Frobenius complements

I have learned that Frobenius complements are characterized (among finite groups) by having a fixed point free complex representation. That is, a finite group $G$ is a Frobenius complement if and only ...
4
votes
1answer
151 views

A small rank linear combination of a small number of elements of a group

This is a version of this question of Klim Efremenko. Let $r>2$ be a natural number, say $r=3$ or $r=10$. Let $G$ be a finite group and $\rho$ be an irreducible complex representation of $G$. We ...
4
votes
2answers
114 views

Measurable representations of semi simple Lie groups

Let $G$ be a semi simple Lie group. I'm particularly interested in $SL(n,\mathbb{R})$. It is proved in I. E. Segal and J. von Neumann, A theorem on unitary representations of semisimple Lie groups, ...
1
vote
0answers
45 views

Approximating eps-homomorphisms

Let $G$ be a finite group. A map $\rho:G\rightarrow U_n$ is called an $\epsilon$-homomorphism if and only if for any $g,h\in G$, we have $||\rho(g)\rho(h)-\rho(gh)||\leq \epsilon$ where the $||$ norm ...
8
votes
1answer
250 views

Division algebras over extension fields / reducibility of $G$-modules

Reformulation of the question (see below for the original question): Let $K$ be an algebraic number field and $D$ a finite-dimensional $K$-division algebra. Is there a description of the field ...
4
votes
1answer
146 views

Questions about the $\mathbf{i}$-trails of Berenstein and Zelevinsky

The $\mathbf{i}$-trails of Berenstein and Zelevinsky was introduced on page 5 (Definition 2.1) in this paper. It is defined as follows. Let $\gamma, \delta \in \mathfrak{h}^*$. Let ${\bf i}=(i_1, ...
2
votes
1answer
128 views

Why are compactly induced representations projective in the category of admissible representations?

I am reading part of Dipendra Prasad's paper found here: http://arxiv.org/pdf/1306.2729v1.pdf. In it (in the middle of page 8) he writes that compactly induced representations are projective. Why is ...
3
votes
1answer
265 views

Jones polynomial of tangles using Temperley-Lieb algbra?

The definition of the Jones polynomial of tangles (a la Reshetikhin and Turaev) uses the space of invariants for $U_q sl_2$ and R-matrices. It seems to me the same thing cane be done in terms of the ...
3
votes
1answer
264 views

Strong Morita equivalence and representation theory

In the context of pure algebra we say that two algebras (in general: rings) $A,B$ are Morita equivalence when there are bimodules $_AP_B,_BQ_A$ such that $P \otimes_B Q \cong _AA_A$ and $Q \otimes_A P ...
2
votes
1answer
239 views

Example of a Frobenius algebra that is not projective over a Frobenius subalgebra

I'd like to know an example of a Frobenius algebra $A$, with a subalgebra $B$ that is itself a Frobenius algebra, such that $A$ is not projective as a left $B$-module. I don't require any ...
1
vote
1answer
136 views

“Generators” for fusion rings

It's a rather obvious idea in the area of fusion rings, but I haven't found a reference yet. Start with the usual rules for a rank n fusion ring $X_i\bigotimes{X_j}=\Sigma_k{T_{ij}}^kX_k$ and ...
15
votes
2answers
423 views

ULU Decomposition of a matrix

Let $g \in GL_n(\mathbb{F}_q)$. Is it true that we can always write $g = u_1lu_2$, where $u_1$ and $u_2$ are upper-triangular and $l$ is lower-triangular? Note that I'm not requiring that the matrices ...
0
votes
0answers
68 views

“Reciprocal” of Schoenberg's theorem

Schoenberg's theorem states that for a (say, countable group) $G$ and any real valued conditionally negative type function $\psi$ on $G$, the function $e^{-t\psi}$ is positive definite, for any ...
6
votes
1answer
368 views

What is the universal property of quotienting a normaliser of the subgroup?

Let $G$ be a group, $H$ a subgroup and $X$ a $G$-set. By taking orbits $X/H = X \times_H 1$ or fixed points $X^H = \mathrm{Hom}_H(1,X)$ we obtain a set on which $H$ acts trivially, and we've destroyed ...
0
votes
0answers
133 views

Comparison of two Chevalley basis

Let $G$ be a connected reductive group over an algebraically closed field and $T$ a maximal torus. Let $H$ be a pseudo-Levi subgroup, say the neutral component of a centralizer of a semisimple element ...
5
votes
0answers
231 views

Torsors and twists of algebraic groups

Let $G/S$ be an affine group scheme. Then the automorphism group of every $G$-torsor over $S$ is a twist of $G$, but it this functor isn't essentially surjective in general (It may be not full nor ...
0
votes
0answers
68 views

kostant partition function vs Haar measure

I am trying to understand the relationship between the Kostant partition function and the Haar measure. Both seem to involve the Vandermonde determinant: $$ \Delta(\theta) = \prod_{i< j} ...