**0**

votes

**2**answers

119 views

### “Diagonalizing” an associative algebra

Consider the associative algebra A with generators $T_i$ and rule $T_i*T_j=\Sigma_kC^{ij}_k*T_k$. Even if it makes no sense for a fusion ring (my momentary pet :-) to change basis it is still possible ...

**3**

votes

**1**answer

146 views

### When does an irreducible unitary real representation remain irreducible after complexifying it?

Consider a unitary real representation of a Lie group $G$ over a real Hilbert space $\mathcal{H}_\mathbb{R}$
\begin{equation}
\rho:G\rightarrow U(\mathcal{H}_{\mathbb{R}})
\end{equation}
Taking the ...

**0**

votes

**0**answers

81 views

### Help finding paper: De Concini, Kac - Quantum Groups at roots of 1

I am looking for a specific paper, that I have found very difficult to trace.
C. De Concini, V. Kac - Quantum Groups at roots of 1
Specifically, the paper is cited as follows (on De Concini's ...

**7**

votes

**0**answers

113 views

### How to show the compatibility between Duflo isomorphism and Harish-Chandra isomorphism for semi-simple Lie algebras?

I was told that the Duflo isomorphism is compatible with the Harish-Chandra isomorphism when the Lie algebra $\mathfrak{g}$ is semi-simple. However I cannot see why this is true. All I can show is ...

**3**

votes

**3**answers

249 views

### Reference request about the representations of the group $PSL_2(\mathbb{F}_q)$

Is there a review/exposition of the representation theory of $PSL_2(\mathbb{F}_q)$ ? Like an enumeration of its irreducible representations and their dimensions as a function of $q$?

**1**

vote

**0**answers

59 views

### Why is the polynomial relating the invariants of a binary polyhedral group fixed by an overgroup?

Let $G$ be a finite subgroup of $\mathrm{SL}(2,\mathbb{C})$ and $N \triangleleft G$ a normal subgroup. Let $x, y, z$ be the fundamental invariants for the standard action of $N$ on $\mathbb{C}^2$, ...

**6**

votes

**0**answers

95 views

### When is a $2$-Calabi–Yau triangulated category the cluster category of a QP?

Keller–Reiten's main theorem in Acyclic Calabi–Yau categories implies that if $\mathcal{C}$ is a $2$-Calabi–Yau (algebraic) triangulated category admitting a cluster-tilting object $T$ such that the ...

**7**

votes

**3**answers

313 views

### Exact sequences of groups and Tannakian formalism

By work of Deligne and others (I am following Deligne-Milne's notes which I just began to read: http://www.jmilne.org/math/xnotes/tc.pdf) we know that a given affine group scheme G can be recovered ...

**4**

votes

**0**answers

89 views

### Geometric interpretation of the Desnanot-Jacobi Identity

Given a square $n\times n$ matrix $M$, let $M_i^j$ denote the $(n-1)\times(n-1)$ matrix obtained from M by omitting the i-th row and j-th column of $M$.
The Desnanot-Jacobi Identity states
...

**1**

vote

**0**answers

47 views

### projective representation of supergroup

In fact, I am not very clear about what I am asking, but I am looking for a concrete example of supergroup which has non-trivial projective representation(some supergroup similar to usual Lie group ...

**1**

vote

**1**answer

113 views

### Generating function for $t$-residues of partitions using Heisenberg + $\hat{sl_t}$ representation theory

Recall that for $t\geq2$, a partition is a $t$-core if none of its hooklengths is divisible by $t$. It is known that the $t$-cores are parametrized by ${\mathbb Z}^{t-1}$. More precisely, let ...

**3**

votes

**0**answers

195 views

### adding a boundary to the finite upper half-plane

Let $\Bbb{F}_q$ be a finite field, let $\delta \in \Bbb{F}_q$ be a non-square, let $\Bbb{F}_{q^2} = \Bbb{F}_q\big( \sqrt{\delta} \big)$ be the corresponding quadratic extension,
and let ...

**3**

votes

**0**answers

101 views

### classifying pairs of idempotent matrices

though classifying pairs of matrices up to simultaneous conjugation is known to be wild, it seems to me a folklore that classifying pairs of idempotent matrices (up to simultaneous conjugation) is ...

**2**

votes

**1**answer

110 views

### Understanding representations of affine Lie algebras

Please reference this paper for notation in this question.
I'm trying to understand two claims made in the above paper (they may be related). First, in the construction of $\mathcal{H}_\lambda$ on ...

**8**

votes

**1**answer

163 views

### finite upper half-plane model for the $\text{GL}_2(\Bbb{F}_q)$ Weil representation

Let $\Bbb{F}_q$ be a finite field with $q$ elements, let $\Bbb{F}_{q^2}$
be its quadratic extension, and consider the finite "upper" half space
${\frak{H}}_q := \Bbb{F}_{q^2} - \Bbb{F}_q$. Apeing a ...

**1**

vote

**1**answer

213 views

### A subalgebra of the Virasoro algebra

Let $L_n$ ($n\in\mathbb{Z}$) and $c$ be the standard generators of the Virasoro algebra ${\rm Vit}$. In the literature one usually considers the involutive authomorphism given by $\tau(L_n)=-L_{-n}$, ...

**1**

vote

**0**answers

87 views

### How does an element $T\left(z\right)$ act on a $\mathcal{U}_{q}\left(\mathcal{L}\mathfrak{sl}_{2}\right)\left[\left[z\right]\right]$-module?

Context
Let $V$ be a 2-dimensional evaluation representation of the quantum loop algebra $\mathcal{U}_{q}\left(\mathcal{L}\mathfrak{sl}_{2}\right)$ with $a=q$. Also, for $m\in\mathbb{Z}$, the ...

**22**

votes

**4**answers

1k views

### Dividing by two in the category of vector spaces

Does every invertible linear map $M$ between $V \oplus V$ and $W \oplus W$ naturally yield an invertible linear map $L$ between $V$ and $W$?
Here "naturally" means "in an $GL(V) \times ...

**0**

votes

**0**answers

67 views

### determinants and principal series representations for $\text{GL}_2 \big( \Bbb{F}_q \big) $

Consider a finite field $\Bbb{F}_q$ and for simplicity let's assume $-1$ is not a square. Let $B$ be the Borel subgroup of $\text{GL}_2 \big( \Bbb{F}_q \big) $, for $i=1, 2$ let $\alpha_i:\Bbb{F}^* ...

**6**

votes

**4**answers

315 views

### Is there a non-explicit characterization of the Specht modules?

It is a basic fact about the symmetric group $S_n$ that its irreducible representations are indexed by partitions of $n$.
My question is, can the association between partitions and irreps be ...

**4**

votes

**0**answers

41 views

### examples of local, nonsemisimple , nonsymmetric hopf algebras

I'm searching for (a class of) examples of Hopf algebras , which have the following properties:
they should be finite dimensional
they should not be semisimple
they should be local
they should ...

**3**

votes

**0**answers

149 views

### regarding the upper half-plane model for the principal series representations of $\text{GL}_2\big( \Bbb{R}\big)$

Let $B$ be the Borel subgroup of $G = \text{GL}_2\big( \Bbb{R}\big)$, let
${\bf \alpha}:B \longrightarrow \Bbb{C}^*$ be a character, and consider the
induced representation $\text{Ind}_B^G ({\bf ...

**6**

votes

**1**answer

307 views

### Quotient of Flag varieties

Let $G=SL_3(\mathbb{C})$ and $X=G/B$ be the associated full flag variety. Fix a non-degenerate symmetric quadratic form $Q$ on $\mathbb{C}^3$. This gives an order $2$ automorphism $F_Q$ of $X$, ...

**9**

votes

**1**answer

218 views

### Why is the A6 preprojective algebra of wild representation type?

As mentioned in the title, I would like to know a proof of the "well known" fact that the A6 preprojective algebra is of wild representation type.
Ideally, I would like to see an explicit ...

**2**

votes

**0**answers

112 views

### Unitary representation of finite-dimensional unitary group

the question is the following. Let n,m be integers, $U(n)$ be the unitary group of $M_n(\mathbb C)$, and $\phi\colon U(n)\to U(m)$ be a continuous group homomorphism, that is moreover irreducible as a ...

**4**

votes

**0**answers

122 views

### Bialgebras with Hopf restricted duals

It is known from the general theory that, given a bialgebra (over a field $k$)
\begin{equation}
\mathcal{B}=(B,\mu,1_B,\Delta,\epsilon)
\end{equation}
the Sweedler's dual $\mathcal{B}^0$ (called also ...

**4**

votes

**0**answers

146 views

### 2-periodic derived equivalence

Let $A$ and $B$ be finite-dimensional algebras with finite global dimension over some field (in fact I am thinking of rational incidence algebras of finite posets).
Suppose we know that $A$ and $B$ ...

**4**

votes

**1**answer

191 views

### Character table of $\mathrm{SL}_2(\mathbb{Z}/p^n\mathbb{Z})$

Is there any reference where I can find the character table of $\mathrm{SL}_2(\mathbb{Z}/p^n\mathbb{Z})$? A simple search in google gave me this paper of Philip C. Kutzko on "The characters of the ...

**0**

votes

**1**answer

119 views

### Equivalence of Lie subalgebras, within a (irreducible) representation

Lie subalgebras inside simple Lie algebras (of type ABCDEFG) have been classified up to equivalence, and linear equivalence (by Dynkin et al). How does one classify embeddings of a Lie algebra h ...

**3**

votes

**0**answers

136 views

### Is there any good survey on the hook length formula and related topics?

I am recently doing some research related to the hook length formula.
The hook formula counts the number of Young tableaux of certain type.
I find there are plenty of research already been done and ...

**1**

vote

**1**answer

190 views

### Infinitely many real roots

Given a non-acyclic quiver without loops with Kac's root system associated. When do we know there are infinitely many real roots?

**2**

votes

**2**answers

236 views

### $t$-analogue of the symmetric power of an additive character over $\Bbb{F}_q^*$

Let $G$ be a finite group and let $f: G \longrightarrow \Bbb{C}$ be any complex-valued function. For integers $k, n \geq 0$, an indeterminant $t$, and $x \in G$ let $f_k(x) := f \big( x^k \big)$ and ...

**6**

votes

**0**answers

189 views

### On a permutation module for GL(n,q)

Let $G=GL(n,q)$ be the general linear group of degree $n$ over the $q$ element field. Let $X$ be the set of full rank $n\times r$ matrices where $1\leq r\leq n$. Then $G$ acts transitively on the ...

**5**

votes

**1**answer

131 views

### Singular/Smooth locus of Schubert variety of the affine grassmannian

Let $G$ be a connected, simply connected, semisimple, complex linear algebraic group with maximal torus $T$ and affine Grassmannian $\mathcal Gr$. It is well known that $\mathcal Gr$ admits a Bruhat ...

**3**

votes

**3**answers

286 views

### A problem with pointwise stabilizer subgroups of fixed-point subspaces II

Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$.
Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$.
Let ...

**0**

votes

**1**answer

94 views

### A problem with pointwise stabilizer subgroups of fixed-point subspaces I

Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$.
Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$.
Let ...

**1**

vote

**1**answer

151 views

### Beilinson-Bernstein localization: $\mathfrak{g}$ action on $G$-equivariant sheaf

I have a few elementary questions related to Beilinson-Bernstein localization.
Let $G$ be a semisimple algebraic group over $\mathbb{C}$ with Lie algebra $\mathfrak{g}$. Consider the setup of ...

**4**

votes

**1**answer

67 views

### Intertwining Operators Associated to Simple Reflections

Let $G$ be a quasi-split reductive group, over a local field, with a Borel subgroup $B=T\cdot N$ and the associated Weyl group $W$. Given a family of induced representations $\pi_s = Ind_B^G \chi\cdot ...

**6**

votes

**2**answers

210 views

### Is an $\mathfrak{sl}_2$-triple determined up to Lie algebra automorphism by the adjoint representation?

Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra, and let $\phi_1:\mathfrak{sl}_2(\mathbb{C})\rightarrow\mathfrak{g}$ and ...

**2**

votes

**0**answers

95 views

### References for the bicategory of ring-bimodule pairs

One of the standard examples of a bicategory is the bicategory of rings (with bimodules as 1-morphisms), which is sometimes denoted $\operatorname{Bim}$ and in other sources $\operatorname{Ring}$ (or ...

**0**

votes

**0**answers

69 views

### Differences between primitive central idempotents and primitive orthogonal idempotents

If we have a complete set of primitive orthogonal idempotents of an algebra $A$, then we can obtain simple modules, indecomposable projective modules, indecomposable injective modules of $A$.
If we ...

**5**

votes

**1**answer

250 views

### Is there a nonabelian finite simple group with Grothendieck ring of multiplicity one?

Let $G$ be a finite group. It admits finitely many irreducible complex representations $H_1, \dots, H_r$ which generate, for $\oplus$ and $\otimes$, the Grothendieck ring $\mathcal{G}(G)$ of $G$ (also ...

**4**

votes

**1**answer

171 views

### Represent matrix immanants using Schur functions

For each irreducible character $\chi^\lambda$ of the symmetric group $S_n$, the immanant of an $n\times n$ square matrix $A$ is defined as
\begin{equation*}
d_\lambda(A) := \sum_{\sigma \in S_n} ...

**1**

vote

**0**answers

49 views

### $\Gamma$ cohomology of principal series

Let $G$ be a noncompact connected real semisimple Lie group with finited center. Let $\Gamma$ be a cocompact discrete subgroup of $G$, and let $P$ be a parabolique subgroup
with Langlands ...

**3**

votes

**1**answer

104 views

### Prescribed spherical representations, symplectic group $Sp(n)$

An irreducible representation $(\pi,V_\pi)$ of a compact group $G$ is called spherical with respect to the pair $(G,K)$, $K$ is closed subgroup of $G$, if $V_\pi$ has a non-zero vector invariant by ...

**3**

votes

**3**answers

180 views

### how to find explicitly given component in a regular representation

Given a finite group $G$ and its irreducible representation $\pi$ I want to find explicit elements of the group algebra $\mathbb{C}[G]$ lying in components of the left regular representation ...

**17**

votes

**1**answer

259 views

### On a drawing in Dixmier's Enveloping Algebras

This image
comes from Dixmier's book, 'Enveloping Algebras' ('Algèbres enveloppantes').
Dixmier writes that
The curves shown on p. XIV have their origin in the study of U(sl(3)).
They are ...

**4**

votes

**1**answer

197 views

### Matrix Elements of Real Representations

I asked this question over at Math.StackExchange and despite having had a bounty on it I did not receive an answer.
Suppose that $G$ is a finite group and we have a unitary irreducible representation ...

**14**

votes

**2**answers

651 views

### Are there any natural differential operators besides $d$?

Let $\lambda = (\lambda_1, \ldots, \lambda_r)$ and $\mu = (\mu_1, \ldots, \mu_r)$ be partitions such that $\mu_j = \lambda_j +1$ for one index $j$ and $\mu_i = \lambda_i$ for all other $i$. Then there ...

**5**

votes

**1**answer

327 views

### Why Jacobson, but not the left (right) maximals individually?

I firstly asked the following question on MathStackExchange, but I did not receive any responses, but a short comment. So, I decided to post it here, hoping to receive answers from experts.
...