# Tagged Questions

**7**

votes

**0**answers

138 views

+50

### Hecke-module structure implicit in definition of automorphic forms in Borel-Jacquet's Corvallis article

Let $G$ be a connected reductive group over a number field $F$, $G_\infty=\prod_{v\mid\infty} G(F_v)$, $\mathbf{A}$ the adèles of $F$, $\mathbf{A}_f$ the finite adèles of $F$. Fix a maximal compact ...

**7**

votes

**1**answer

189 views

### If $G$ is compact, $H \leq G$ open, $V$ an irreducible $H$-rep, is $\text{Ind}_H^G$ semisimple?

Let $G$ be a compact group, $H$ a normal open subgroup, and $K$ a $p$-adic field (so that not all $G$-reps with coefficients in $K$ are semisimple). Let $V$ be a finite-dimensional topological ...

**2**

votes

**0**answers

64 views

### Update on list of open problems for Cherednik/Symplectic Reflection Algebras

Background:
There are two lists of open problems about Cherednik or Symplectic Reflection Algebras from 2007: Ian Gordon's Problems, Chapter 9 in Symplectic Reflection Algebras, and Ginzburg & ...

**3**

votes

**1**answer

57 views

### $C^\infty$-vectors in general representations of Lie groups on locally convex spaces

This question is related to
this one. Let $G$ be a real Lie group (I should emphasize I only care about ordinary Lie groups, not Lie groups modeled on locally convex spaces or anything like that). In ...

**1**

vote

**1**answer

196 views

### labeling state vectors in representation space of a simple lie algebra

Given a simple lie algebra (over ${\mathbb C}$ or ${\mathbb R}$). What is the number
of operators such that their eigenvalues sufficiently label all state vectors in the algebra's representation ...

**2**

votes

**0**answers

66 views

### $G$-invariant part of products of determinants of minors

Let $G = SL_n$; then for any tuple $\lambda$ such that $\sum \lambda_i = n$, define $f_\lambda(g)$ as the product of the determinants of successive minors of lengths $\lambda_i$ of $g$ (e.g. for ...

**1**

vote

**0**answers

86 views

### References for crystal bases and Demazure modules in representation theory

I was wondering what are some standard general references/books/survey articles about: (1) crystal bases, and string parameterizations and (2) Demazure modules, and Schubert varieties (containing ...

**17**

votes

**6**answers

2k views

### Applications of group theory to math. biology (pharmacology) ?

Are there applications of group theory (take it broadly: representation theory, Lie algs., q-groups, whatever ... ) to math. biology ?
I am in particular interested about applications to pharmacology ...

**2**

votes

**2**answers

163 views

### Example of linearization for GIT

Take a vector space $V$ (finite dimensional, over the complex numbers), let $G=SL(V)$. The group $G$ acts on $\mathbb{P}V$ and we can linearize its action to an action on the line bundle ...

**3**

votes

**1**answer

146 views

### Condition on a Hopf operad for tensor product in the base categoy to be a (categorical) coproduct for algebras

A Hopf operad will be an operad endowed with a coproduct $P(n) \longrightarrow P(n) \otimes P(n)$ which is compatible in the obvious sens with operad laws (no more structure is assumed a priori. ...

**2**

votes

**1**answer

97 views

### Invariants of polynomial algebras on infinite G-modules

Is there a reasonable characterization of the ring of $G$-invariants
($G$ - a finite dimensional simple Lie group) in the polynomial algebra
on direct sum of infinite number of irreducible ...

**4**

votes

**1**answer

211 views

### Calculation with weights of $E_6$

Question: Consider the complex simple Lie group $E_6$. Let $\lambda_1$ and $\lambda_6$ be the fundamental weights defining the $27$-dimensional representation $V$ and $V^*$, resp. Consider the complex ...

**6**

votes

**1**answer

145 views

### Integrals of representations over geodesics

Let $G$ be a compact, connected Lie group and $\rho$ any of its irreducible, unitary representations. If $\gamma:S^1\to G$ is an injective homomorphism (a periodic geodesic passing through the ...

**2**

votes

**1**answer

85 views

### Symmetric invariants of a Schur Module

Let $V\cong\mathbb C^n$ be a complex vector space of dimension $n$. Let $\lambda\in\mathbb Z^r$ be a generalized integer partition $\lambda_1\ge\cdots\ge\lambda_r$ with $r\le n$. Denote by $\mathbb ...

**9**

votes

**1**answer

375 views

### Number of standard Young tableaux with fixed corner entry

For a partition $\lambda=(\lambda_1\geq \lambda_2\geq\ldots\geq \lambda_k)$ of $n$, let the set of standard Young tableau of shape $\lambda$ be denoted by $SYT(\lambda)$ with boxes at $(i,j)$ denoted ...

**12**

votes

**0**answers

221 views

### Adams Operations on $K$-theory and $R(G)$

One of the slickest things to happen to topology was the proof of the Hopf invariant one using Adams operations in $K$-theory. The general idea is that the ring $K(X)$ admits operations $\psi^k$ that ...

**4**

votes

**1**answer

442 views

### Is the Duflo polynomial conjecture open?

Let $G/K$ be a symmetric space. Let
$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ be a Cartan decomposition,
with the odd part $\mathfrak{p}$. It is well known that the algebra of invariant
...

**-4**

votes

**1**answer

57 views

### How to find matrix representations of a boolean algebra? [closed]

Given a boolean algebra with a finite number of elements {a, b, c, ...}, and the usual operations: $\cup, \cap, \neg$.
How to find matrix representations of the elements such that:
boolean $\cup$ ...

**5**

votes

**2**answers

247 views

### When is/isn't the monoidal unit compact projective?

I am interested in developing intuition for when the monoidal unit in a closed monoidal abelian category is or isn't compact projective. As such, my question is not looking for a yes/no answer, but ...

**0**

votes

**0**answers

71 views

### Is it possible to factor the tensor of two representations of a group $G$ through $G \otimes G$? [closed]

With direct sum there is a simple way to factor the representation through $G \oplus G$:
$$ G \to G \oplus G \to GL(V \oplus W) $$
where the first map is the diagonal and the second is through the ...

**6**

votes

**0**answers

125 views

### Sum over growing Young tableaux

Let $\lambda_0,\lambda_1,\lambda_2,\lambda_3,\ldots$ be a sequence of Young diagrams, such that each successive diagram is obtained from the prior by the addition of one box (don't forget that the row ...

**11**

votes

**1**answer

497 views

### Counting representations of $k[x,y]$ when $k$ is finite

$\newcommand{\GFq}{\mathbf F_q}$
Let $r_n(q)$ denote the number of isomorphism classes of $n$-dimensional modules of the $\GFq$-algebra $\GFq[x,y]$. Is it known whether there exists a polynomial ...

**9**

votes

**1**answer

357 views

### Vector bundles, Higgs bundles and the Langlands program

This question is somewhat vaguely structured. But, I hope someone can make it more precise (or) it is indeed possible to answer it in the form that I am stating it.
Background : I recently chanced ...

**0**

votes

**0**answers

92 views

### Non-semisimple Lie algebra tensors

Let $\mathfrak{L}$ be a non-semisimple Lie algebra. Let $T_i$ be its generators. As usual, define the structure constants $C_{ij}^k$ by $[T_i,T_j]=C_{ij}^kT_k$ and the metric tensor $g_{jm}$ by ...

**3**

votes

**0**answers

100 views

### Parshin's buildings for higher local fields

What is the status of the theory of buildings for higher local fields?
I know that there are some papers of Parshin, in which he describes some examples, like $PGL_2$ and $PGL_3$ over ...

**1**

vote

**1**answer

67 views

### Is it necessary for $\pi:H\to U(\mathcal{H}_{\pi})$ to be a homomorphism in order for $\text{ind}_H^G\pi$ to be weakly continuous?

Suppose $G$ is a locally compact group and $H$ is an open subgroup for simplicity. Further suppose $\pi$ is a representation of $H$ on some Hilbert space $\mathcal{H}_{\pi}$, i.e. $\pi(h)$ is unitary ...

**5**

votes

**2**answers

348 views

### What is natural about the well-known bijection between conjugacy classes and irreps of a symmetric group?

Symmetric groups possess a well-known bijection between conjugacy classes and irreducible representations. More precisely, both sets are indexed by Young diagrams.
Question: To what extent is this ...

**3**

votes

**0**answers

66 views

### A canonical algebra of type $(2,2,r)$ is derived equivalent to a path algebra of type $\tilde{D}_{r+2}$ (references)

According to several articles I could find, a canonical algebra of type $(2,2,r)$ is derived equivalent to a path algebra of type $\tilde{D}_{r+2}$, where $r \geq 2$.
I don't know how to obtain this ...

**13**

votes

**1**answer

399 views

### Categorifying the equality of product and coproduct of symmetric functions

Littlewood-Richardson coefficients are both multiplicities of $GL_n$ tensor products, and of restrictions of $GL_{m+n}$ representations to $GL_m \times GL_n$. I want to turn this equality of numbers ...

**1**

vote

**1**answer

102 views

### Fibers of the Bott-Samelson Resolution of Schubert Varieties

Is there an explicit (perhaps visual) description of the fibers of the Bott-Samelson Resolutions of Schubert Varieties? Let's fix $G$ to be $GL_n(\mathbb{C})$.
Also, how would the answer to the ...

**0**

votes

**2**answers

746 views

### On a technical fact used in the proof of density of smooth vectors in a representation

My question concerns a fact used in Knapp's Representation Theory of Semisimple Groups implicitly in the proof of Lemma 3.13, which is crucial for the proof of density of smooth vectors in a ...

**8**

votes

**0**answers

129 views

### Representations of orthogonal groups vs representations of reflection groups

Let $V$ be a finite dimensional inner product space and $O(V)$ the orthogonal group of $V$.
Let $G$ be a (say, finite) reflection group on $V$, regarded as a subgroup of $O(V)$ ($G< O(V)$.) Let
us ...

**4**

votes

**3**answers

153 views

### Questions on constructions of supercuspidal representations

To my knowledge, usually there are two ways to construct supercuspidal representations over p-adic fields. The first is via theory of types (for GL(n) and classical groups), notably by Bushnell, ...

**4**

votes

**0**answers

91 views

### Higman's Criterion

Higman's Criterion is a fundamental result in the representation theory of a finite group $G$ over a field $k$ of characteristic $p>0$. One version is the following (I hope the notation is standard ...

**2**

votes

**1**answer

458 views

### 1D TQFT in Freed-Hopkins-Lurie-Teleman

In the first section of Freed-Hopkins-Lurie-Teleman they construct a one-dimensional Topological Quantum Field Theory.
$F(\circ_+)$ is a vector space and $F(\circ_-)$ is the dual.
$F(\circ-\circ)$ ...

**3**

votes

**1**answer

392 views

### Special automorphisms of extraspecial groups

Let $G$ be an extraspecial group of order $p^{2r+1}$ (where $p$ is an odd prime), and let $V$ be a faithful representation of $G$. Consider the normal subgroup $H$ of $Aut(G)$ consisting of all ...

**5**

votes

**1**answer

168 views

### Quadratic Casimir of fundamental irreps of simply-laced Lie algebras

I have the following question, motivated by the expression for the character of level 1 highest weight integrable representations of simply-laced affine algebras (in terms of the string function). It ...

**4**

votes

**2**answers

389 views

### What does Freed-Hopkins-Teleman say about finite groups?

The third in the "Loop groups and twisted K-theory" series by FHT treats compact Lie groups without any connectedness assumptions. I am trying to unwind what Theorem 2 of that paper (available here, ...

**5**

votes

**0**answers

164 views

### Which L-functions are not “Langlands-Shahidi L-functions”?

The Langlands-Shahidi method, among other things, obtains certain L-functions from the constant term of Eisenstein series attached to so-called $(G,M)$ pairs, where $G$ is a reductive group, $M$ a ...

**3**

votes

**0**answers

119 views

### Invariants of the symmetric group

Let $V_\lambda$ be an irreducible representation of the symmetric group $S_n$ as usual labeled by parition $\lambda$ of $n.$
Question.
Is there any general information about the algebra of ...

**1**

vote

**1**answer

115 views

### classification of irreducible finite dimensional representation of affine hecke algebra of type A

Let $H_{n}$ be the affine Hecke algebra with parameter q, where q is not root of unity.
The classification of irreducible finite dimensional representations has been given by Kazhdan-Lusztig in terms ...

**2**

votes

**1**answer

145 views

### Computing tangent spaces of resolutions to Slodowy slices

This question is about (a special case of) the varieties discussed here: Does the preimage of the Slodowy slice in $T^*G/P$ have a name?.
Let $G = SL_n(\mathbb{C}), \mathfrak{g} = ...

**2**

votes

**0**answers

109 views

### Cohomology of Bott-Samelson varieties?

How is the cohomology of Bott-Samelson varieties (desingularizations of Schubert Varieties ) usually calculated? Let's fix the Lie group to be $GL_n(\mathbb{C})$ or $SL_n(\mathbb{C})$ here.
Is there ...

**9**

votes

**1**answer

232 views

### Decomposition of $\mathrm{O}(n)$-modules coming from differential geometry

Let $V$ be a $n$-dimensional real vector space equipped with a positively definite scalar product $g$ and let $\mathrm{O}(n)$ be the automorphism group of $(V,g)$. View $V^{\otimes k}$ as a ...

**1**

vote

**2**answers

341 views

### Chinese Remainder Theorem backwards

I have the following situation, that is much alike the Chinese Remainder Theorem. Let $\phi_d(\alpha)$ be the $d^{th}$ cyclotomic polynomial in the variable $\alpha$ (I'm not specifying the ...

**5**

votes

**0**answers

107 views

### Generators for invariant tensors of lie algebras

EDITED FOR (hopeful) CLARITY:
For a simple Lie algebra $\mathfrak{g}$ (over $\mathbb{C}$) and its adjoint group G, the $G$-invariant polynomials on $\mathfrak{g}$ are linear combinations of products ...

**3**

votes

**1**answer

163 views

### Lie group GL(4) representation decomposition

Let $V$ be the defining representation of $GL(4,\mathbb C)\to GL(V)$ with $V=\mathbb{C}^4$.
Let $Ext\;V$ be the exterior square of $V$ which is a 6-dim repsentation.
My question: How does $$V\otimes ...

**9**

votes

**1**answer

227 views

### Representation varieties of 3-manifold groups in SL(n,C)

I am looking at the variety of representations of the fundamental group of a hyperbolic 3-manifold into $SL(n,C)$:
$$Hom(\pi_1M, SL(n,{\mathbb C}))$$
It is known that volume and Chern-Simons ...

**3**

votes

**1**answer

64 views

### Intersection of Maximal Left Ideals with Finite Dimensional Quotient

Let $\Gamma$ be a finitely generated group and let $A=\mathbb{C}[\Gamma]$ be the corresponding group algebra over $\mathbb{C}$. Let $X$ be the set of all maximal left ideals of $A$ and let $X_0=\{I ...

**3**

votes

**1**answer

99 views

### Is there a left-adjoint to the restriction of comodules?

If $f\colon C \to D$ is a homomorphism of coalgebras and $\rho\colon V \to V\otimes C$ is a $C$-comodule, then $(1\otimes f)\rho\colon V \to V\otimes D$ is the comodule restricted to $D$. In e.g. ...