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

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Evaluation modules of $U_q(L(sl_2))$

Let $a \in \mathbb{C}^{\times}$, $r \in N$. Let $W = V_q(r)$ be the $r$-dimensional irreducible type 1 representation of $U_q(gl_2(\mathbb{C}))$. In the usual basis $\{v_0, \ldots, v_r\}$, the action ...
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179 views

Is the assignment of a root system to a complex semisimple Lie algebra functorial?

As described here, we have a category of root systems, where a morphism from a root system $\Phi$ in a Euclidean space $E$ to a root system $\Phi'$ in $E'$ is given by a linear map $f: E \to E'$ such ...
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34 views

Is it possible for a quantum group algebra $U_{q}\left(\mathfrak{g}\right)$ to have a diagonal universal $R$-matrix?

I am writing a research paper and have shown that in the special case when a quantum group algebra $U_{q}\left(\mathfrak{g}\right)$ with the quantum group parameter $q$ not a root of unity has a ...
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251 views

Understanding the purely formal part of the sheaf theoretic (cohomological) framework for representation theory

By now I have the impression that many statements in representation theory can be phrased a lot more elegantly using cohomological language. Therefore I'm trying to understand a bit better the sheaf ...
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46 views

Action of longest element of Weyl group on zero weight space

Let: $G$ be a real semisimple Lie group; $\rho$ be an irreducible representation of $G$ on a finite-dimensional real vector space; $A$ be a "Cartan subspace" of $G$ (a Lie subalgebra which is ...
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1answer
115 views

Maps between symmetric powers of the natural module for $SL_2 (k)$ in prime characteristic

Let $G=SL_2(k)$ considered as a linear algebraic group over an algebraically closed field of prime characteristic. Let $E$ be the natural module for $G$ and denote by $S^r (E)$ its $r-$th symmetric ...
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77 views

a modular character problem [on hold]

Let $B\in$Bl$(G|D)$ and suppose that $\sigma\in$Aut$(G)$ fixes every $\chi\in$Irr$(B)$. If $d\in D$, show that $d$ and $d^\sigma$ are $G$-conjugate. It is a problem from Navarro's book "characters and ...
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159 views

Artin conjecture on L-functions

Artin conjecture on Artin $L$-functions asserts that the Artin $L$-function $L(\rho,s)$ of a non-trivial irreducible representation $\rho$ of the Galois group $\Gamma$ of a number field admits ...
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112 views

Corepresentations of Tensor Products of Hopf Algebras

Given two cosemisimple Hopf algebras $H,G$ over ${\mathbb C}$, denote their usual (not braided) tensor product by $G \otimes H$. What conditions do we need to impose on the Hopf algebras to ensure ...
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99 views

Projectives in the category of modular representations of Lie algebras

Let $\mathfrak{g}$ be a semi-simple Lie algebra (eg. $\mathfrak{g} = \mathfrak{sl}_n$), defined over an algebraically closed field $\textbf{k}$ with characteristic $p >> 0$. The center $Z(U\...
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103 views

“Nice” basis for highest-weight irreducible module of a simple Lie algebra

Let $\mathfrak{g}$ be a simple complex Lie algebra, $\mathfrak{h}$ a Cartan subalgebra, $\Phi \subset \mathfrak{h}^*$ the associated root system, $\Sigma = \{\sigma_i : i\in I\}$ a basis of simple ...
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118 views

Spectral decomposition on GL(n)

If $\Delta_1, \ldots, \Delta_{n-1}$ are a basis of the ring of commuting bi-$SL(n,R)$-invariant differential operators, $L_0^2=L_0^2(SL(n,Z)\backslash SL(n,R))$ is the space of cuspidal automorphic ...
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94 views

When does an irreducible G-module admit an invariant quadratic form of signature (n,n+1)

Let $G$ be a connected real reductive Lie group and $V$ be a finite dimensional real irreducible $G$-module. When does $V$ admit an invariant non-degenerate quadratic form of signature $(n,n+1)$? I ...
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112 views

Singular Locus of a Schubert variety

I am trying to compute the singular locus of the schubert variety $X_w$ in $G_{2,7}$ where $w=(4,7) \in I_{2,7}$. Following the notation in the book "The Grassmannian Variety: Geometric and ...
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77 views

Bounding Schur polynomials of a particular shape

Consider Schur polynomials $s_\lambda$ with $\lambda = (2m, m, m, \ldots, m, 0)$ and $\ell(\lambda) = n$ (that is, $\lambda$ has $n$ rows). Here $m \gg n$, which, for the sake of concreteness, let's ...
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417 views

When can a finite subgroup of $GL(2n,\mathbb{R})$ be viewed as a subgroup of $GL(n,\mathbb{C})$?

A finite group acting on a complex vector space of dimension $n$ can be seen as acting on a real vector space of dimension $2n$ just by forgetting the complex structure of the space. My question is, ...
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273 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 ...
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244 views

Simple groups and irreducible characters of degree 3

I have posted this question on mathstack echange but did not get any answer. It mam trying my luck here. The only simple finite groups admitting an irreducible character of degree 3 are $\mathfrak{A}...
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1answer
147 views

A representation of Spin(9,1)

Let $Spin(9,1)$ denote the universal (double) cover of $SO(9,1)$. $Spin(9,1)$ acts linearly on $\mathbb{R}^{16}$ (see e.g. p.29 here https://arxiv.org/pdf/math/0105155v4.pdf ). Consider the induced ...
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65 views

Decomposition into irreducible components of a representation of $Spin(9)$

It is well known that the group $Spin(9)$ acts linearly on the vector space $\mathbb{R}^{16}$ (see for example "Spinors and calibrations" by R. Harvey). Consider the induced representation of $Spin(9)...
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192 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 $\...
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378 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 ...
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98 views

On an inequality about asymptotics of Whittaker functions

I'm reading Wallach's paper 'Asymptotic expansions of generalized matrix entries of representations of real reductive groups'(Lecture Notes in Math., 1024,287–369) and got confused by one statement ...
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169 views

representation of a group and its center

(I asked the following question at StackExchange but received no answer.) Let $G$ be a finite group and let $Z(G)$ be its center. Let $C=\mathrm{Rep}(G)$ be the category of finite dimensional ...
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1answer
204 views

Maximal Coset representative for the Weyl group of a Parabolic

Let $G=SL_n$ and let $P_i$ be a maximal parabolic corresponding to a simple root say $\alpha_i$. Let $W_{P_i}$ be the Weyl group of $P_i$. Is there an efficient way to compute the longest coset ...
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152 views

'Noether normalization' for finite group schemes

Throughout let $p$ be a prime, and let $k$ be a field of characteristic $p$. Let $G$ be a compact Lie group. Such a $G$ can always be embedded as a closed subgroup of $SU(n)$ for some $n$. This ...
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155 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 ...
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167 views

Associated vector bundles and Characteristic Classes

Assume that $P\to M$ is a principal $G$-bundle where $G$ is some (compact) Matrix group. Let $\rho\colon G \to \operatorname{Gl}(\mathbb{R}^n)$ be the tautological representation and $\rho^\prime\...
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35 views

Mathematical Definition of $n$-Brouillin Zone [duplicate]

I am having trouble finding a mathematical definition of the Brouillin zone beyond the first, which are basically the Voronoi cells or Wigner-Seitz cells. We could imagine the set of point closer to ...
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128 views

Are there any unitary matrices which satisfy the Yang-Baxter equation which are universal for quantum computation?

Let $H$ be a finite dimensional hilbert space. Let $L:H\otimes H\rightarrow H\otimes H$ be a unitary transformation. Then the equation $$(L\otimes I)(I\otimes L)(L\otimes I)=(I\otimes L)(L\otimes I)(I\...
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56 views

Comodules of the $B,C$ and $D$ series quantum groups

In Section 11.5 of Klimyk and Schmudgen's book on quantum groups, explicit presentations of the isomorphism classes of comodules of ${\cal O}(GL_q(N))$ are given in terms of its "quantum minors". In ...
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129 views

Decay of Fourier coefficients for compact Lie groups

Let $G$ be a compact Lie group, $G^\natural$ the space of conjugacy classes in $G$ with the natural pushforward of $G$'s Haar measure. Let $f\in L^2(G^\natural)$. Then the Peter–Weyl Theorem tells us ...
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Action is determined by a braiding

Let $H$ be a bialgebra over $\mathbb{C}$. Suppose that $V$ is a left $H$-comodule and $W$ is an $H$-module. Then we can defined map $\Psi$ by \begin{align} & \Psi: V \otimes W \to W \otimes V, \\ &...
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Realisation of the preprojective algebras as $F(\Delta)/T$ over some quasi-hereditary algebra

Let $A$ be the Auslander algebra of $K[x]/(x^n)$ for some $n \geq 2$, which is quasi-hereditary with some characteristic tilting module $T$. Dlab and Ringel showed in their paper "The Module ...
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177 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 \...
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448 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 ...
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97 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 ...
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1answer
45 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 ...
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1answer
91 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 ...
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1answer
210 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 ...
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322 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{...
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106 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-...
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242 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$(V//G)$...
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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)...
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154 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 ...
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175 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 $x^i$ ...
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110 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)?
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59 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)}....
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50 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 ...
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98 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 ...