**10**

votes

**1**answer

184 views

### equivariant stable class of quaternionic Hopf fibration in RO(G)-degrees of ADE-type

Does the quaternionic Hopf fibration possibly represent a non-torsion element in the $G$-equivariant stable homotopy groups of spheres, for $G$ a finite subgroup of $SO(3)$ and in RO(G)-degree being ...

**8**

votes

**0**answers

196 views

### $\mathcal{M}(\mathcal{D}_X)$ and $\mathcal{M}^r(\mathcal{D}_X)$ have natural tensor category structures?

Write $\mathcal{M}^\ell(\mathcal{D}_{X/S}) = \mathcal{M}(\mathcal{D}_{X/S})$ for the category of left $\mathcal{D}$-modules over $X$ and $\mathcal{M}^r(\mathcal{D}_{X/S})$ for the category of right ...

**3**

votes

**0**answers

110 views

### When is a given quiver algebra a hopf algebra?

Given a finite dimensional selfinjective quiver algebra A over a finite field (or more generally an arbitrary field). Whats the best way to check if the algebra A has a Hopf algebra structure or not? ...

**5**

votes

**2**answers

216 views

### Representations of orthogonal groups over the field of two elements

I am looking for some references on modular representation theory of the orthogonal groups $O_{2n+1}(2)$, $O_{2n}^{+}(2)$, or $O_{2n}^{-}(2)$ over $\mathbb{F}_2$.

**2**

votes

**0**answers

96 views

### Center of $U_q(sl_3)$ and $U_q(sl_4)$

In the book a guide to quantum groups, page 285, the center of $U_q(g)$ is described in Theorem 9.1.6. The center of $U_q(sl_2)$ is computed explicitly in Example 9.1.7. I tried to compute the center ...

**3**

votes

**0**answers

105 views

### Integer-matrix representation of a commutative ring

Consider a commutative ring $x_ix_j = N_{ij}^k x_k$, where $N_{ij}^k \in\{0,1,2,3,\cdots\}$, and $\{x_i\}$ is a finite set. (This is actually a fusion ring and $x_i$ are simple objects.)
How to find ...

**1**

vote

**1**answer

126 views

### Equivariant polynomial maps

Let $V$ be a complex vector spaces and assume that a compact group G acts linearly on $V$. Then look at the $G$-equivariant polynomial maps from $V$ to $V$. Denote this by $Mor_G(V,V)$. In the case ...

**5**

votes

**0**answers

218 views

### Recurrence Formula for Zernike polynomials

I'm not sure if this is research level, so if this result is known, please excuse the intrusion. I am trying to find a relation between solutions of the Laplacian equation in $4$ dimensions and those ...

**2**

votes

**1**answer

87 views

### Number of cluster variables

In the paper cluster algebras and quantum affine algebras, Section 13.5, it is said that when $\mathfrak{g}$ is of type $A_2$ and $\ell=2$, then the corresponding cluster algebra $\mathscr{A}_2$ for ...

**2**

votes

**1**answer

189 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 ...

**20**

votes

**1**answer

600 views

### Which philosophy for reductive groups?

I am just beginning to look further into trace formulas and automorphic forms in a quite general setting. For long I have noticed that the natural assumption on the groupe $G$ we work on is to be ...

**2**

votes

**1**answer

106 views

### Conceptual explanation for multiplicativity of theta generalization of extreme characters of U(infty)

A character of $U(\infty)$ is a continuous, positive definite class function $\chi: U(\infty) \longrightarrow \mathbb{C}$, which is normalized by $\chi(e) = 1$. Observe that the set of characters of ...

**4**

votes

**0**answers

101 views

### Intertwiners, $\text{dim}_\mathbb{C}(\text{End}_G \mathbb{C}\{X\})$? [closed]

Let $\mathbb{F}$ be a finite field and let $G = SL_2(\mathbb{F})$. The group $G$ acts linearly on the $2$-dimensional vector space $\mathbb{F}^2$ and fixes the origin $0 \in \mathbb{F}^2$. Hence, $G$ ...

**0**

votes

**0**answers

58 views

### Homomorphisms of quantum spaces

Suppose we have a look at the Hopf $*$-algebra $U_q(sl(2))$ and the Hopf $*$-algebra $A_q(\widetilde{S}^3)$ introduced in the paper: http://arxiv.org/pdf/q-alg/9605017v1.pdf. I want to find a relation ...

**3**

votes

**1**answer

100 views

### Finding a semigroup that maximizes the trace of a sum of matrices

Let $H$ be a finite semigroup containing $n$ elements from a compact group $G$. I am trying to solve
$$\max_{h_i,\ h_j\ \in\ H} \operatorname{tr} \sum_{i,\ j\ \leq\ n} \rho(h_i^{-1} h_j)(A_j ...

**6**

votes

**0**answers

169 views

### What's the name of the cohomology class associated to a projective representation?

Suppose $\rho : G \to PGL_n(k)$ is a projective representation of a group $G$ over a field $k$. It's classical that the obstruction to lifting this to a linear representation $G \to GL_n(k)$ is a ...

**9**

votes

**0**answers

261 views

### Vanishing theorems in algebraic geometry and representation theory

Garland proved vanishing theorems for the cohomology of a discrete subgroup with coefficients in a finite dimensional complex representation. As I understand it, Casselman reproved them using the ...

**3**

votes

**2**answers

107 views

### Covering derivations of a quotient algebra

Let $(\mathcal{A},+,·)$ an algebra and $\mathcal{I}$ an ideal of $\mathcal{A}$.
Is easy to check that if $D\in Der(\mathcal{A})$ with $D(\mathcal{I})\subseteq I$ then $D$ induces a derivation $D_I$ ...

**9**

votes

**1**answer

428 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 ...

**5**

votes

**2**answers

124 views

### Relationship between the Lie functor applied to a Lie group action, and the fundamental vector field mapping?

Let $M$ be a smooth manifold, and $G$ a Lie group with Lie algebra $\mathfrak{g}$. The Lie algebra of the diffeomorphism group of $M$ is the Lie algebra of vector fields on $M$; that is ...

**5**

votes

**3**answers

392 views

### Why is the trace of the Casimir on the irrep of a semisimple algebra nonzero?

A crucial step in the "purely algebraic" proof of Weyl's semisimplicity theorem is that the Casimir element $C\in U\mathfrak{g}$ acts by nonzero scalars on a nontrivial irrep $V$. However, at least ...

**2**

votes

**1**answer

128 views

### Representations of the algebra of odd quantum spheres

I read the article by Dijkhuizen and Noumi (http://arxiv.org/pdf/q-alg/9605017v1.pdf). Here they describe in section $4$ what the algebra of functions on the total space of a family of quantum ...

**0**

votes

**1**answer

159 views

### Reference Request: Irreducibles of the regular representation of the permutation group is absolutely irreducible

I am writing a paper(physics) where I am using the fact that the irreducible's of the regular representations of the permutation group are absolutely irreducible in the following sense.
If $V$ is an ...

**3**

votes

**1**answer

184 views

### Is the Veronese variety “enough” to describe all the $SL(V)$-orbits in $\mathbb{P}(\textrm{Sym}^dV)$?

I apologise in advance if the question will look ridicolous to experienced eyes: in this case a good reference will be enough to clarify my doubts.
Let $V$ be a complex vector space of dimension $n$, ...

**11**

votes

**2**answers

328 views

### Quotient rule, differential operator on a localization is well-defined, underlying geometry?

Using the quotient rule, we obtain that the notion of differential operator on a localization is well-defined:$$\mathcal{D}_A(B_f) \cong \mathcal{D}_A(B)_f.$$Here, $B$ is a commutative $A$-algebra, ...

**8**

votes

**1**answer

186 views

### Pair of square matrices related by traces formulas

Let $A$ and $B$ be two $n\times n$ matrices over $\mathbb{C}$. Assume that for every $k\geq 1$ it holds $tr(A^k) = tr(B^{2k-1})$. What can we say about the possible eigenvalues of $A$ and of $B$? How ...

**4**

votes

**0**answers

74 views

### For any $f \in B$ which is not nilpotent, the set consisting of powers of $f$ is a multiplicative set in $\mathcal{D}_A(B)$? [closed]

Let $B$ be a commutative $A$-algebra. Let $\mathcal{D}_A(B)$ be the ring of differential operators of $B$ over $A$. Does it follow that for any $f \in B$ which is not nilpotent, the set consisting of ...

**1**

vote

**0**answers

60 views

### Partial orders on tabloids

Let $n \in \mathbb{N}$ and let $\lambda \vdash n$, a partition of $n$. By a $\lambda$-tabloid I mean a row-tabloid of shape $\lambda$. There is a well-known order on the set of $\lambda$-tabloids, ...

**3**

votes

**0**answers

81 views

### Branches of 3j symbols

Question
Is there a quick way to identify the branches in a 3J symbol?
Context
I need to compute Wigner 3J symbols/Clebsch–Gordan coefficients,
$$
\begin{pmatrix}
\ell_1 &\ell_2 &\ell_3\\
...

**6**

votes

**2**answers

272 views

### Faithful projective representations of symmetric groups

This is a reference request.
Do you know where I can find the dimensions of the faithful projective representations of $S_n$ and $A_n$ for $n\ge 5$?
Thank you in advance.

**2**

votes

**0**answers

87 views

### Prove that a Verma module is projective only if its highest weight is dominant?

Let $\mathfrak{g}$ be a finite-dimensional semisimple Lie algebra over $\mathbb{C}$ with a fixed Cartan subalgebra $\mathfrak{h}$ and a fixed system of simple roots. It is stated in Exercise 3.11 of ...

**9**

votes

**1**answer

227 views

### Does it follow that any element of $J(A)$ is nilpotent?

Let $A[x]$ be the algebra of polynomials with coefficients in a $k$-algebra $A$. Assume that, for any simple $A[x]$-module $M$, we have $\text{End}_{A[x]} M = k$. Does it follow that any element of ...

**7**

votes

**3**answers

524 views

### Beyond Brauer's theorem

Brauer's classic theorem states that any character of a finite group can be expressed as a linear combination with integer coefficients of characters induced by linear characters of p-elementary ...

**2**

votes

**1**answer

100 views

### Maximal possible dimension of abelian Lie subalgebra of Heisenberg Lie algebra of dimension $2n+1$? [closed]

Fix $n \in \mathbb{N}$, and let $\mathfrak{h}_n$ denote the Heisenberg Lie algebra of dimension $2n+1$ (over any given field $k$). Namely, $\mathfrak{h}_n$ is the Lie algebra with basis $x_1, \dots, ...

**10**

votes

**1**answer

167 views

### Is $F_{f, c, \ell}$ a $G$-harmonic polynomial?

Let $G \subset \text{GL}_n(\mathbb{C})$ be a finite subgroup. The group $G$ acts naturally on $\mathbb{C}^1[\mathbb{C}^n]$ the space of degree $1$ homogeneous polynomials in $x_1, \dots, x_n$, i..e, ...

**1**

vote

**0**answers

132 views

### Is there a brute force method for determining irreducible representations?

Suppose I have some groups $G_1$, $G_2$, $G_3$, etc... Then the direct product is given by $G = G_1 \times G_2 \times G_3 \ldots$
I know that the sub-representations of a reducible representation ...

**4**

votes

**1**answer

87 views

### Is there a nice form for the Frobenius characteristic of a border shape character?

Let $\chi^V$ be the character of a border strip Specht module, i.e. a Specht
module for a skew tableau that contains no $2 \times 2$ square. I know that
the Frobenius characteristic of $\chi^V$ is ...

**3**

votes

**0**answers

115 views

### Seeking an unpublished manuscript by Tetsuro Okuyama

Several papers in representation theory attribute the notion of relatively projective modules to Tetsuro Okuyama's manuscript "A generalization of projective covers of modules over finite group ...

**6**

votes

**1**answer

139 views

### Two quivers, finitely many nonisomorphic representations of $\mathbb{C}Q$

Consider the following two quivers:
...

**0**

votes

**0**answers

59 views

### Tau functions for the KP and Toda lattice hierarchies

A statement, which is known to be true can be vaguely stated as "the tau function for the KP and Toda hierarchies are the same". I would just like to know exactly what it means as it is not obvious ...

**3**

votes

**0**answers

96 views

### line bundle on affine grassmannian and central extension

Let $G$ be a connected reductive group over $\mathbb{C}$, let $Gr$ be the affine grassmannian of $G$. On $Gr$, we know that there is a canonical line bundle $L$ (the generator of $Pic(Gr)$).
Now ...

**2**

votes

**0**answers

65 views

### Alternating elements in free graded-commutative algebras

It is classical that every alternating polynomial is (uniquely) the product of a symmetric polynomial with the Vandermonde polynomial, in particular the alternating polynomials are a free rank-one ...

**1**

vote

**0**answers

68 views

### Irreducible representations of quantum affine algebras

The finite dimensional simple modules of quantum affine algebras are parameterized by Drinfeld polynomials. Some other modules of quantum affine algebras are studied in the paper. Some infinite ...

**5**

votes

**2**answers

173 views

### Center of quantum affine algebras

Are there some references about the center of quantum affine algebras? I searched on google and only find the paper. In particular, what is the center of $U_q(\widehat{\mathfrak{sl}_2})$. Thank you ...

**3**

votes

**0**answers

81 views

### Quasi-split subgroup

Let $G$ be a split symplectic group over a number field $K$, and let $T \subset G$ be a maximal torus defined over $K$ (not necessarily a split torus). The long roots of $T$ form a ...

**2**

votes

**1**answer

87 views

### Why is the ker-hull-topology on $Irr(A)$ is the discrete topology?

Let $A$ be a C$^*$-algebra. Let $Irr(A)=\{[\pi]: \pi$ is an irreducible representation of A}, here is $\rho\in [\pi]$ if there is an unitary operator $V:H_{\pi}\to H_{\rho}$ such that ...

**5**

votes

**0**answers

98 views

### (Double) Crystal reflection operators on SSYTs

I am not that familiar with the language of crystals, but this is what I know:
Let $SSYT(\lambda, \mu)$ be the set of semi-standard Young tableaux with shape $\lambda$ and weight $\mu$.
There are ...

**5**

votes

**1**answer

164 views

### Regular functions on nilpotent orbits and their covers

Let $G$ be a complex semisimple algebraic group with Lie algebra $\mathfrak{g}$.
In 1989 McGovern described the structure (as $G$-module) of the ring of regular functions on a finite cover of the ...

**2**

votes

**0**answers

219 views

### Properties of Higman's group

The infinite group of Higman which has no finite quotient is given by the presentation (with 4 generators and 4 relations):
$$
G = \langle a_i, i \in \mathbb{Z}/4\mathbb{Z} \mid a_ia_{i+1 \,(\text{mod ...

**9**

votes

**0**answers

113 views

### Failure of surjectivity in Hotta-Springer specialization: examples for special unipotents?

Last weekend's workshop on Springer theory and its generalizations at UMass demonstrated how far the subject has expanded over four decades, but the original set-up for the Springer correspondence ...