**2**

votes

**1**answer

116 views

### Tensor and symmetric invariants of Symmetric group

For the action of $S_n$ on $\mathbb C^n$ the elementary symmetric polynomials generate the ring of polynomial invariants. What are the generators for the action of $S_n$ on $\mathbb C^n \otimes \...

**9**

votes

**0**answers

329 views

### Why should Algebraic Geometers and Representation Theorists care about Geometric Complexity Theory?

Geometric Complexity Theory has demonstrated that Complexity Theorists should care about Algebraic Geometry and Representation Theory, but, why should Algebraic Geometers and Representation Theorists ...

**4**

votes

**1**answer

57 views

### Involutions and Little Adjoint Representations of Simple Algebras

In what follows I'm going to use $V_{\theta_s}$ for the little adjoint representation af a Lie algebra i.e. the representation associated with the highest short rooth $\theta_s$.
Is easy to see that ...

**1**

vote

**1**answer

44 views

### Reference request: compatibility conditions of four versions of Yetter-Drinfeld modules

There are four versions of compatibility conditions of Yetter-Drinfeld modules (left-left, left-right, right-left, right-right) in the article. Are there some references which derive these ...

**1**

vote

**1**answer

82 views

### How to show that Yetter-Drinfeld condition is equialent to $\Psi$ is a braiding

Let $H$ be a Hopf algebra and $V$ a right $H$-module and right $H$-comodule. The module $V$ is a Yetter-Drinfeld module over $H$ if and only if
\begin{align}
( v \triangleleft h_{(2)} )_{(0)} \otimes ...

**2**

votes

**0**answers

78 views

### Irreducible representations of Sp(2)

I am looking for an explicit description of the finite dimensional irreducible representations of the classical Lie group $\text{Sp}(2) = \{A\in M_2(\mathbb{H})\,|\,A\overline{A}^T = I\}$.
I can ...

**6**

votes

**1**answer

297 views

### Why does the Bogolyubov transformation work? - In language of Clifford Algebras?

Letting the standard Clifford algebra of dimension $2k$ be denoted by $Cl_{2k}$, let's denote the corresponding complex Clifford algebra via $$\mathbb{C}l_{2k}\equiv Cl_{2k}\otimes_{\mathbb{R}}\mathbb{...

**1**

vote

**0**answers

44 views

### Quiver folding and maximal green sequences

The technique of quiver folding (please see Folding by Automorphisms) can be used to prove statements about non-simply laced quivers (i.e. valued quivers) when they are already known in the simply-...

**7**

votes

**1**answer

232 views

### Is the dimension of $V//G$ always the same as the dimension of $V^*//G$ ?

I would like to know whether there is an example of a reductive algebraic group $G$ (say, over the complex numbers $\mathbb{C}$) and a finite dimensional representation $V$ of $G$ such that dim$(V//G)$...

**1**

vote

**0**answers

48 views

### Is $T(V)$ a Yetter-Drinfeld module over $T(V^* \otimes V)$?

Let $V$ be a vector space and $V^*$ the dual vector space. Let $T(V)$ be the tensor algebra of $V$. Is there some action $T(V^* \otimes V) \otimes T(V) \to T(V)$ and coaction $T(V) \to T(V^* \otimes V)...

**4**

votes

**1**answer

117 views

### Sum of Young symmetrisers of a given shape

Preliminaries and notation:
Let $n\in \mathbb{Z}_{>0}$ and $\lambda=(\lambda_1,\lambda_2,\dots,\lambda_s)\vdash n$ be a partition. Given a Young diagram of shape $\lambda$, we can associate it ...

**4**

votes

**0**answers

171 views

### Enumerating a class of polynomials

How many equivalence classes of $\Bbb F_2[x,y]$ polynomials with $x$ degree $n_x$ and $y$ degree $n_y$ are there such that each $y^i$ coefficient (polynomial in $\Bbb Z[x]$) is distinct and $x^i$ ...

**2**

votes

**2**answers

108 views

### Unitary representations of SO(1,4) and SO(2,3)

Where can I find details about the irreducible unitary representations of SO(1,4) and SO(2,3)?

**2**

votes

**1**answer

51 views

### Compatibility conditions for Yetter-Drinfeld modules

In the paper, page 28, Definition 4.2.1, the compatibility condition for a Yetter-Drinfeld module over $H$ is
$$
h_{(1)}.v_{(-1)} \otimes h_{(2)}.v_{(0)} = (h_{(1)}.v)_{(-1)}h_{(2)} \otimes (h_{(1)}....

**1**

vote

**0**answers

46 views

### What's the symplectic form preserved by a rational representation of a semisimple group

Let $\mathbb{H}_1$, $\mathbb{H}_2$ be two quaternion algebras over $\mathbb{Q}$ and $G_1 = SL_1(\mathbb{H}_1)$, $G_2 =SL_1(\mathbb{H}_2)$.
Over $\mathbb{C}$, $G_1\sim G_2 \sim SL_2(\mathbb{C})$. I'm ...

**4**

votes

**1**answer

88 views

### When is the category of Gorenstein projective $R$-modules Frobenius?

Let $R$ be a ring (associative with unit, but not necessarily commutative, and definitely not necessarily Noetherian.) Then the category $\operatorname{GP}(R)$ consists of those $R$-modules having a ...

**3**

votes

**2**answers

132 views

### Do the irreducible modules of this finite group preserve a tensor product structure?

I am interested in a particular group $G$, where
$$ (A_4\times C_\ell) \lhd G \lhd S_4 \times D_\ell$$
Here, $C_\ell$ is cyclic, $D_\ell$ is dihedral of order $2\ell$, and the two inclusions both have ...

**1**

vote

**0**answers

73 views

### Is $T(V) \rtimes T(V^* \otimes V)$ a bialgebra?

Let $V$ be a vector space and $V^*$ the dual vector space. Let $T(V)$ be the tensor algebra of $V$.
The algebras $T(V)$ and $T(V^* \otimes V)$ are bialgebras. I am trying to find some bialgebra ...

**2**

votes

**0**answers

61 views

### Center of affine W-algebras

Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra and $k$ a complex number. Denote by $\hat{\mathfrak{g}}$ the corresponding affine Lie algebra ($\hat{\mathfrak{g}}=\mathfrak{g}((t)...

**1**

vote

**1**answer

45 views

### Factorizability of Subquotients of Principal Series Representations

Fix number field $F$, its ring of adeles $\mathbb{A}$, a "nice" algebraic group defined over $F$ (at least reductive but for my purposes I can assume simple and simply connected) and a parabolic ...

**4**

votes

**1**answer

257 views

### Do irreducible characters form a closed set?

A character on a discrete group $\Gamma$ is a conjugation-invariant function $\tau$ which is of positive
type, and is normalized so that $\tau(e) = 1$, where $e$ is the identity element of $\Gamma$. A ...

**4**

votes

**0**answers

115 views

### Characterizing the RSK corespondance

The Robinson-Schensted-Knuth correspondence is an algorithm which takes as input a word $w$ on the alphabet $\{1,\dots,n\}$ of length $k$ and returns a pair of a tableau $P(w)$ and a standard tableau $...

**2**

votes

**0**answers

142 views

### Irreducible representations containing simple actions of $\mathrm{SL}(2,\mathbb{C})$

Let $G$ be a complex semisimple Lie group and let $\rho: G \longrightarrow \mathrm{SL}(n,\mathbb{C})$ be a faithful irreducible representation of $G$ with $n \geq 3$. Suppose that $G$ contains a copy ...

**6**

votes

**1**answer

166 views

### Real-valued character in Block with cyclic defect has at most two constituents modulo $p$

Let $G$ be a finite group and let $(K,R,k)$ be a $p$-modular system (large enough for $G$ etc.) and consider a block algebra $B \subseteq RG$ with cyclic defect group.
My question is about the ...

**6**

votes

**0**answers

111 views

### Software for explicit computations in representations of classical Lie algebras

I'm pretty sure many a mathematician has longed for such a tool but I wasn't able to find such a question here, so here we go.
Is there, by any chance, an existing package or program that allows one ...

**3**

votes

**0**answers

196 views

### Equivariant sheaves over affine schemes

Let $k$ be a field, let $G$ be a linear algebraic group over $k$ and
let $A$ be a commutative $k$-algebra which is acted on by $G$.
We say that an $A$-module $M$ is a $(G,A)$-module if it satisfies ...

**4**

votes

**1**answer

158 views

### What is known about the decomposition of $Sym(Sym^3(V))$ into irreducibles?

The representation $\text{Sym}(\text{Sym}^3(V))$ of $\text{GL}(V)$ decomposes into a direct sum of $S_{\lambda}(V)$, where the $S_{\lambda}$ are Schur functors. What is know about this decomposition?
...

**3**

votes

**0**answers

129 views

### Invariant functions on the dual Lie algebra

Let $G$ be a real Lie group and $\mathfrak{g}$ the corresponding Lie algebra. Let $\mathfrak{g}^*$ be the dual of the Lie algebra. Then we have the coadjoint action of $G$ on $\mathfrak{g}^*$.
...

**2**

votes

**0**answers

60 views

### Can we write an element in a super Grassmannian as a pair of matrices?

Super Grassmannians are introduced by Manin, see for example.
Elements in a grassmannian can be written as matrices, see for example.
Can we write an element in a super Grassmannian as a pair of ...

**2**

votes

**0**answers

34 views

### Characteristics of $c$-vectors of acyclic cluster algebras

In Speyer and Thomas's work, Acyclic Cluster Algebras Revisited the characteristics of $c$-vectors of cluster algebras with the $B$-matrix of the initial seed acyclic are given in Theorem 1.4. Do we ...

**4**

votes

**1**answer

170 views

### Tilting modules in positive characteristic

Consider the category of finite-dimensional representations for the algebraic group $\mathrm{SL}(n)$ in characteristic $p$. I know very little about this but am told there is a highest weight category ...

**7**

votes

**1**answer

239 views

### Is it known whether every symmetric pair of finite groups of Lie type is a Gelfand pair?

A pair of groups $(G,H)$ is called a symmetric pair if $H$ is the group of fixed points of an involutive automorphism of $G$, for example $(GL(2n,\mathbb{F}_q),Sp(2n,\mathbb{F_q}))$ is a symmetric ...

**6**

votes

**0**answers

115 views

### Are the weight spaces of indecomposable $U_q\mathfrak{sl}(2)$-modules at most 2-dimensional?

This is a follow up of this question.
Let $U_q\mathfrak{sl}(2)$ be Lusztig's integral form of the quantized enveloping algebra of $\mathfrak{sl}_2$, specialised at $q$ a root of unity. This is an ...

**-1**

votes

**1**answer

119 views

### Crystal basis for quantum groups and Lie algebras

Lie $g$ be a finite dimensional complex simple Lie algebra and $U_q(g)$ the corresponding quantum group, where $q$ is not a root of unity. Every simple finite dimensional $g$-module is of the form $V(\...

**1**

vote

**0**answers

117 views

### Classification of finite subgroup of $PGSp_4(\mathbb{C})$

Is there a classification of the finite subgroups of $PGSp_4(\mathbb{C})$?

**11**

votes

**1**answer

1k views

### Why is this character expression an integer?

Let $\gamma$ be an $n$-dimensional complex representation of a finite group $G$ with character $\chi$ and let $e=c_0, c_1, ..., c_{\ell}$ be a set of conjugacy class representatives for $G$. In the ...

**1**

vote

**0**answers

63 views

### Representation equivalent lattices

Suppose $G$ is a absolutely almost simple algebraic groups over a number field $K$. Let $\Gamma_1$ and $\Gamma_2$ be two lattices in $G(K)$. Then $\Gamma_1$ and $\Gamma_2$ are said to be ...

**0**

votes

**0**answers

101 views

### Normalizer of non-split tori

Let $\mathbb{G}$ be a connected reductive group over $\mathbb{C}$. Let $G:=\mathbb{G}(\mathbb{C}(\!(t)\!))$. Let $T$ be a maximal torus in $G$.
Question: What do we know about the normalizer $N_G(T)$...

**4**

votes

**0**answers

73 views

### Expressing knot polynomials as casimirs

I always wondered why one writhe unit (read: colored with the irrep of a quantum Lie algebra and evaluated as Reshetikhin-Turaev invariant thereof) is essentially the quadratic casimir of that irrep. (...

**0**

votes

**0**answers

32 views

### Simple modules of quantum toroidal algebras

Many properties of quantum toroidal algebras are similar to quantum affine algebras. Every simple module of a quantum affine algebra of rank $n$ corresponds to an $n$-tuple of Drinfeld polynomials.
...

**1**

vote

**2**answers

162 views

### Motivations of cross product

Let $H$ be a bialgebra and $B$ an $H$-module. The cross product $B \rtimes H$ of $B$ and $H$ is $B \otimes H$ as a vector space and the multiplication in $B \rtimes H$ is defined as follows: $(a \...

**5**

votes

**0**answers

79 views

### non unitary representations

A paper by M.L. Whippman in Rep. Math. Phys. 5 (1974), 81, mentions at the bottom of the second page ''the possible occurrence of non-unitary representations that arise when reducing a direct product ...

**2**

votes

**1**answer

130 views

### Product of Bruhat Cells

Fix a $(B,N)$ pair (Tits system) of a semisimple Lie group $G$. Let $u$ and $v$ be two Weyl group elements such that $l(uv)=l(u)+l(v)$. It is known that $BuvB=(BuB)(BvB)$ (see for example Humphreys's ...

**0**

votes

**1**answer

102 views

### $Ext$ functor over a product of groups

Let $G_1$ and $G_2$ be two groups (of some kind, e.g. finite groups).
Let $M_1, N_1$ be $G_1$-modules, and $M_2, N_2$ be $G_2$ modules, always with coefficients in $\mathbb{C}$.
Write $G = G_1 \...

**5**

votes

**2**answers

134 views

### Multiplication in universal enveloping algebra in terms of PBW isomorphism

Let $\mathfrak g$ be a Lie algebra. Consider the multiplication map $m:\mathfrak g\otimes U(\mathfrak g)\to U(\mathfrak g)$ and $i:S(\mathfrak g)\to U(\mathfrak g)$ -- Poincare-Birkhoff-Witt ...

**4**

votes

**0**answers

123 views

### $p$-forms acting on spinors

I'm trying to understand the paper by Atiyah, Hitchin and Singer called: ''Self-duality in four dimensional Riemannian geometry", available here.
I'm stuck at the point where it explains how the $p$-...

**4**

votes

**0**answers

96 views

### Partial formality of A-infinity structure implies formality

Let $A$ be a (finite dimensional, unital, associative) $k$-algebra, where $k$ is a (algebraically closed) field. Let $M$ be a (finite dimensional) $A$-module. Then, it is known that $\operatorname{Ext}...

**2**

votes

**2**answers

135 views

### Realization of irreducible $\mathfrak{S}_d$-modules and the representation theory of Lie algebra

Let $n$ be a positive integer. It is well-known that a method to realize irreducible $\mathfrak{S}_d$-modules is to construct the so-called Specht modules $S^{\mu}$ which are submodules in the so-...

**1**

vote

**0**answers

94 views

### Decompositon of the Euler class in the ideal generated by Weyl-invariant polynomials

Let $G$ be a complex reductive Lie group, $B$ be a Borel subgroup, $T\subset B$ be a maximal torus, $W$ be the Weyl group. Then the space $X:=G/B$ is a complex manifold of dimension $n$, denote by $\...

**16**

votes

**0**answers

269 views

### What are the indecomposable $U_q\mathfrak{sl}(2)$-modules?

Let $\mathfrak g=\mathfrak{sl}(2)$.
Let $\zeta$ be a primitive root of unity of even order. Say $\zeta=e^{2\pi i/6}$, for concreteness.
Let $U_q\mathfrak g$ be Lusztig's integral form of the ...