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

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5
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382 views

What is a good introduction to branching rules in representation theory?

I'm looking for a book or introductory article, that explains branching rules in representation theory of real Lie groups. When a Lie group has a set of irreducible representations, I'd like to know ...
3
votes
0answers
67 views

Is the Symplectic Schur algebra a 0-faithful cover of the Brauer algebra?

The symplectic Schur algebra, $Sp_{2n}$ and the Brauer algebra, $B_r(n)$, are in Schur-Weyl duality over an algebraically closed field of characteristic $p$ (this is due to Doty et. al.). The ...
10
votes
1answer
280 views

Asymptotic Weyl Character Formula

Let $G$ be a complex semi-simple group along with a chosen pair of opposite Borel subgroups (so we get all the root-theoretic data we need). Let $\lambda$ be a dominant weight, and let $V(\lambda)$ be ...
11
votes
3answers
674 views

Does the Alternating group of degree $n>7$ have exactly one irreducible character of degree $n-1$?

We know that the alternating group of degree $n>7$ has an irreducible character of degree $n-1$. The latter number is the smallest nontrivial one for each the alternating group has an irreducible ...
3
votes
1answer
251 views

A question on algebraic loop groops

Setup: Let $\mathcal{K}=\mathbb{C}((t))$, $\mathcal{O}:= \mathbb{C}[[t]]$ and $G$ be a reductive algebraic group (over $\mathbb{C}$). Let further $\mathcal{K}_n$ denote the $\mathcal{O}$-ideal in ...
1
vote
1answer
165 views

Do Auslander-Reiten quivers coincide with the McKay quivers for arbitrary subgroups of GL(2,C)?

It is a theorem of Auslander that if $G< GL(2,\mathbb C)$ is a finite subgroup without pseudo-reflections, then the Auslander-Reiten quiver of $K[x,y]^G$ coincides with the McKay quiver of $G$ with ...
6
votes
0answers
195 views

Alternative source to Drozd's book on finite dimensional algebras

I am trying to learn classic representation theory of finite dimensional algebras. My main source is the book "finite dimensional algebras" by Drozd and Kirichenko. I did not have too much trouble ...
4
votes
1answer
207 views

Is $(G,K)$ a strong Gelfand pair?

Let $F$ be a $p$-adic field with ring of integers $\mathcal{O}$. When $G={\rm GL}_n$, it is a classical result that $(G(F),G(\mathcal{O}))$ is a Gelfand pair. Is it actually a strong Gelfand pair? I ...
-1
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1answer
153 views

irreducible Classical Lie algebras [closed]

which submodule of FG-module of a lie algebra $L$ will be determined I want to check that how we can find out a classical lie algebra like $D_4$ and $E_6$ are irreducible?
0
votes
1answer
101 views

spectrum of an induced algebra

Let $G$ be a reductive group defined over an algebraically closed field $k$ and $B$ be a fixed Borel subgroup of $G$. Suppose $X=Spec(R)$ is an affine scheme with $B$ rationally acting on it; hence ...
4
votes
1answer
176 views

Towards a quantum version of Schur's orthogonality relations

This is taken from Timmermann's Invitation to Quantum Groups and Duality. Hi folks I am struggling a little with a small calculation in the above text. I will just get right into it. Lemma 3.2.5 ...
13
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325 views

Combinatorics of Quantum Schubert Polynomials

Let $S_n$ be the symmetric group. Let $s_i$ denote the adjacent transposition $(i \ i+1)$. For any permutation $w\in S_n$, an expression $w=s_{i_1}s_{i_2}\cdots s_{i_p}$ of minimal possible length is ...
2
votes
0answers
107 views

Criteria for a finite-dimensional $k$-Algebra to be basic and elementary

I have the following question: Suppose, I have a finite dimensional $k$-Algebra $A$ over an arbitrary field $k$ and a finite dimensional module $M$ that is a generator-cogenerator of mod-$A$. I'm ...
0
votes
0answers
140 views

special values of L-functions cohen-lenstra heuristic

I found some lecture notes on links between number theory and random permutations. It was difficult to follow: The notes start with an interesting fact, whose proof I've asked on Math.StackExchange: ...
4
votes
2answers
377 views

Linearisation of a group

If $G$ is a connected Lie group acting on a vector $\mathbb{C}$-space $V$ then it is well known that the algebra of invariants $\mathbb{C}[V]^G$ coincides with the algebra of invariants ...
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0answers
127 views

Is this a pure monomorphism?

Let $M$ be a representation of a quiver $Q=(V, E)$ by $R$-modules. By $M^{+}$ we mean a representation of $Q^{op}$ with $M^{+}(v)=\mathrm{Hom}(M(v), \frac{Q}{Z})$. One can easily see that there is ...
3
votes
2answers
166 views

Moving Between Weight Spaces in Highest-Weight Representations

Let $G$ be a connected, simply-connected complex semisimple linear algebraic group with Lie algebra $\frak{g}$. Fix a maximal torus $T\subseteq G$ and let $\Delta\subseteq Hom(T,\mathbb{C}^*)$ be the ...
0
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2answers
239 views

Decomposition of $SU(3)$ representation $6\times 15$ into irreducibles?

The 6 and 15 dimensional representations of $SU(3)$ are irreducible. The 90 dimensional tensor product representation $6\times 15$ decomposes into a sum of irreducible representations. What factors ...
6
votes
3answers
294 views

A Hausdorff abelian group with no character?

Pontryagin Duality for locally compact abliean groups gives plenty of continuous (unitary) characters $\chi : A \to \mathbb{R} / \mathbb{Z}$, but if we do not assume local compactness, can anything be ...
2
votes
0answers
108 views

A generalization of Macdonald functions?

I am interested in finding a set of functions $f(z_1,\cdots ,z_k;q,\,t)$, conjecturally polynomials, which depend on two parameters $(q,t)$ and an integer $k$, and are orthogonal under the following ...
2
votes
0answers
76 views

minimal graded projective resolution

Suppose we are given a positively graded $k$ algebra $\bigoplus_{i\ge 0}A_i$ such that $A_0$ has finite global dimension. Furthermore all $A_i$ are finite dimensional and $A$ is generated in degree 0 ...
4
votes
1answer
98 views

Extension of characters of abelian locally compact groups

Let $G$ be an abelian locally compact group and $H$ be its closed subgroup. It is known from Pontryagin duality theory that every unitary character of $H$ can be extended to $G$. I think this is true ...
7
votes
2answers
232 views

Are torus knot groups linear?

The fundamental group $T(p,q)$ of the complement of a $(p,q)$-torus knot (in $S^3$) admits the presentation $\langle a, b \mid a^p=b^q \rangle $. Is $T(p,q)$ linear, i.e., is there a faithful ...
2
votes
3answers
529 views

Are all (possibly infinite dimensional) irreducible representations of a commutative algebra one-dimensional?

If $A$ is a commutative algebra over an algebraically closed field $k$, and $\rho:A \rightarrow End(V)$ is an irreducible representation of $A$ (where, a priori, $V$ may be infinite dimensional), can ...
4
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0answers
129 views

Scaling-Invariant Orbits of Semisimple Group Representations

Let $G$ be a connected, simply-connected complex semisimple linear algebraic group, and let $V$ be a finite-dimensional complex $G$-module. Note that if $V$ is the adjoint representation of $G$, then ...
5
votes
1answer
195 views

When are orbits of semisimple group representations closed?

Let $G$ be a connected, simply-connected complex semisimple linear algebraic group, and let $V$ be a finite-dimensional complex $G$-module. Is there a nice description of those $v\in V$ for which the ...
7
votes
0answers
219 views

Higher-dimensional generalization of Pink's theorem

Pink's theorem in the title of the question refers to the main theorem of Pink's paper "Compact Subgroups of Linear Algebraic Groups" that appeared in Journal of Algebra (206) in 1998. It essentially ...
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votes
0answers
84 views

Use of “abstract tensors” for trivalent graphs

Preamble: For knots, a necessary and sufficient set of "graphic rules" for using Kauffmans abstract tensors was given by Turaev. The lowest dimensional solutions, interpreting these abstract tensors ...
3
votes
0answers
73 views

The Tangent Bundle of the Space of CR Structures on S^(2n+1)

Let $M$ be a smooth compact $n$-manifold without boundary, $g$ some choice of Riemannian metric on $M$, and $\omega_g$ the volume form gotten from $g$. Say you're interested in finding extrema for ...
6
votes
1answer
146 views

regular semisimple elements on spherical varieties

Let $(G,H_1)$ and $(G,H_2)$ be spherical pairs (i.e. $G$ is a reductive group, $H_i$ are its closed subgroups and the Borel subgroup $B$ of $G$ has a finite number of orbits on $G/H_i$). What can ...
3
votes
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39 views

points with small U stabilizer on a spherical variety

Let $(G,H)$ be a spherical pair (i.e. $G$ is a reductive group, $H$ is a closed subgroup and the Borel subgroup $B$ of $G$ has a finite number of orbits on $G/H$). Let $U$ be the unipotent radical of ...
14
votes
3answers
700 views

Algebraic Groups in Characteristic p

It is well-known that Lie groups are, under nice conditions, essentially determined by their Lie-algebras. What's the corresponding statement for algebraic groups over fields of finite characteristic? ...
3
votes
0answers
90 views

parametrization of irreducible finite dimensional representation of Weil group

Let $F$ be a p-adic field, with p a prime denoting the residue field characteristic. Let $\mathcal{W}_F$ be the Weil group. In the local Langlands correspondence for $GL(n,F)$, it is important to know ...
2
votes
0answers
113 views

Correct definition of locally algebraic parabolic induction of a locally algebraic character

Let $L$ be a finite extension of $\mathbf{Q}_p$ and $G$ the group of $L$-points of a split connected reductive group $\mathbf{G}$ over $L$, $T$ the $L$-points of a split maximal torus in $\mathbf{G}$, ...
2
votes
1answer
328 views

On a result due to Zelevinskii

In his paper on the p-adic analogue of the Kazhdan-Lusztig hypothesis (Functional Analysis and Its Applications 15.2 (1981): 83-92), Zelevinskii proves a combinatorial proposition (outlined in Section ...
4
votes
1answer
262 views

When does a group action on a k-algebra induce an algebraic action on the spectrum?

This question arose from my last question, which I considered answered - from the comments, however, it is obvious that the answer is only complete in characteristic zero, and I am trying to ...
4
votes
1answer
217 views

Jacquet module for Lie algebras?

Let $G$ be a reductive group and let $P$ be a parabolic subgroup with Levi $L$ and unipotent radical $N$. Let $F$ be a finite or local non-Archimedean field. The Jacquet Functor is a functor from ...
2
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0answers
143 views

The fundamental in the tensor square of a complex representation of $SO(N)$

I would like to figure out whether there is an irreducible complex (in the sense non-self-conjugate) representation of a group $SO(N)$, $N>2$, whose tensor square contains the fundamental ...
2
votes
1answer
158 views

a question about the semidihedral group?

My question is simple: If a group $G$ has the same character table with the semidihedral group $SD_{2n}$, are $G$ and $SD_{2n}$ isomorphic ?
1
vote
1answer
414 views

Para-Complexification of Lie Groups

Let $G$ be a real Lie group. Then the complexification $G_\mathbb{C}$ of $G$ is the unique complex Lie group equipped with a map $φ:G\to G_\mathbb{C}$ such that any map $G\to H$ where $H$ is a ...
12
votes
3answers
710 views

Why do we need a $G$-universe?

Let $G$ be a compact Lie group. Before defining $G$-prespectra, we have to define a $G$-universe $\mathcal U$. Question: Why do we need a $G$-universe? A $G$-universe is defined to be a countably ...
3
votes
1answer
390 views

Wedderburn decomposition of $D_{5}$

This is crossposted from MSE. The question: Find the Wedderburn decomposition of $D_{5},$ the dihedral group of order 10, over the field $\mathbb{F}_{3}.$ I have shown that the irreducible ...
7
votes
0answers
182 views

The space-time dimension of the N-superstring theory?

Let $\mathfrak{W}$ be the Lie algebra generated by $d_{n} = ie^{in\theta}\frac{d}{d\theta}$ and $\mathfrak{Vir} = \mathfrak{W} \oplus C \mathbb{C}$ its central extension: $$ ...
5
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214 views

Which de Rham representations are trianguline?

Let $K/\mathbf{Q}_p$ be a finite extension, and let $V$ be an $n$-dimensional $\overline{\mathbf{Q}_p}$-vector space with a continuous action of $G_K$. Suppose $V$ is de Rham, so potentially ...
5
votes
0answers
163 views

Are there exactly solvable CFTs?

I am wondering if there are CFTs such that n-point correlation functions in them of the fields (may be the primaries or of some notion of twist fields) is exactly known. Are there such? Aren't ...
15
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248 views

Quasi-classical limit of representation theory

I am looking for a good reference on a general phenomenon of quasi-classical limit in representation theory, which relates "large" representations to measures on (co-adjoint orbits of) the associated ...
11
votes
1answer
770 views

Is this error in this paper of Langlands fixable?

The FQS criterion for the Virasoro algebra was discovered by Friedan, Qiu and Shenker (1), but the mathematicians found their proof insufficient, so that, FQS (2) and Langlands (3), published in the ...
5
votes
2answers
353 views

Conjugacy classes of PGL(3,Z)

We know that every $2\times 2$ matrix in $PGL(2, \mathbb{Z})$ of order $3$ is conjugate to the matrix $$ \left( \begin{array}{cc} 1 & -1 \\ 1 & 0 \end{array} \right) $$. I am interested in ...
0
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0answers
67 views

Decomposability of representations of Clifford algebras

As far as I have understood, any representation of a semisimple Lie algebra is decomposable, i.e., it can be brought on block-diagonal form (by some similarity transformation). But what about ...
5
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217 views

generalized Koszul algebras

Let A be a positively graded algebra such that A_0 has finite global dimension (see "On a common generalization of Koszul duality and tilting equivalence" by Dag Madsen, arXiv.org > math > ...