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

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9
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389 views

Vector bundles, Higgs bundles and the Langlands program

This question is somewhat vaguely structured. But, I hope someone can make it more precise (or) it is indeed possible to answer it in the form that I am stating it.  Background : I recently chanced ...
4
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109 views

Parshin's buildings for higher local fields

What is the status of the theory of buildings for higher local fields? I know that there are some papers of Parshin, in which he describes some examples, like $PGL_2$ and $PGL_3$ over ...
1
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1answer
71 views

Is it necessary for $\pi:H\to U(\mathcal{H}_{\pi})$ to be a homomorphism in order for $\text{ind}_H^G\pi$ to be weakly continuous?

Suppose $G$ is a locally compact group and $H$ is an open subgroup for simplicity. Further suppose $\pi$ is a representation of $H$ on some Hilbert space $\mathcal{H}_{\pi}$, i.e. $\pi(h)$ is unitary ...
3
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71 views

A canonical algebra of type $(2,2,r)$ is derived equivalent to a path algebra of type $\tilde{D}_{r+2}$ (references)

According to several articles I could find, a canonical algebra of type $(2,2,r)$ is derived equivalent to a path algebra of type $\tilde{D}_{r+2}$, where $r \geq 2$. I don't know how to obtain this ...
4
votes
1answer
213 views

Calculation with weights of $E_6$

Question: Consider the complex simple Lie group $E_6$. Let $\lambda_1$ and $\lambda_6$ be the fundamental weights defining the $27$-dimensional representation $V$ and $V^*$, resp. Consider the complex ...
2
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1answer
118 views

Fibers of the Bott-Samelson Resolution of Schubert Varieties

Is there an explicit (perhaps visual) description of the fibers of the Bott-Samelson Resolutions of Schubert Varieties? Let's fix $G$ to be $GL_n(\mathbb{C})$. Also, how would the answer to the ...
0
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0answers
98 views

Non-semisimple Lie algebra tensors

Let $\mathfrak{L}$ be a non-semisimple Lie algebra. Let $T_i$ be its generators. As usual, define the structure constants $C_{ij}^k$ by $[T_i,T_j]=C_{ij}^kT_k$ and the metric tensor $g_{jm}$ by ...
8
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134 views

Representations of orthogonal groups vs representations of reflection groups

Let $V$ be a finite dimensional inner product space and $O(V)$ the orthogonal group of $V$. Let $G$ be a (say, finite) reflection group on $V$, regarded as a subgroup of $O(V)$ ($G< O(V)$.) Let us ...
4
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95 views

Higman's Criterion

Higman's Criterion is a fundamental result in the representation theory of a finite group $G$ over a field $k$ of characteristic $p>0$. One version is the following (I hope the notation is standard ...
2
votes
1answer
469 views

1D TQFT in Freed-Hopkins-Lurie-Teleman

In the first section of Freed-Hopkins-Lurie-Teleman they construct a one-dimensional Topological Quantum Field Theory. $F(\circ_+)$ is a vector space and $F(\circ_-)$ is the dual. $F(\circ-\circ)$ ...
4
votes
2answers
396 views

What does Freed-Hopkins-Teleman say about finite groups?

The third in the "Loop groups and twisted K-theory" series by FHT treats compact Lie groups without any connectedness assumptions. I am trying to unwind what Theorem 2 of that paper (available here, ...
3
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123 views

Invariants of the symmetric group

Let $V_\lambda$ be an irreducible representation of the symmetric group $S_n$ as usual labeled by parition $\lambda$ of $n.$ Question. Is there any general information about the algebra of ...
5
votes
2answers
354 views

What is natural about the well-known bijection between conjugacy classes and irreps of a symmetric group?

Symmetric groups possess a well-known bijection between conjugacy classes and irreducible representations. More precisely, both sets are indexed by Young diagrams. Question: To what extent is this ...
5
votes
2answers
291 views

When is/isn't the monoidal unit compact projective?

I am interested in developing intuition for when the monoidal unit in a closed monoidal abelian category is or isn't compact projective. As such, my question is not looking for a yes/no answer, but ...
5
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0answers
240 views

Which L-functions are not “Langlands-Shahidi L-functions”?

The Langlands-Shahidi method, among other things, obtains certain L-functions from the constant term of Eisenstein series attached to so-called $(G,M)$ pairs, where $G$ is a reductive group, $M$ a ...
4
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0answers
126 views

Cohomology of Bott-Samelson varieties?

How is the cohomology of Bott-Samelson varieties (desingularizations of Schubert Varieties ) usually calculated? Let's fix the Lie group to be $GL_n(\mathbb{C})$ or $SL_n(\mathbb{C})$ here. Is there ...
9
votes
1answer
239 views

Decomposition of $\mathrm{O}(n)$-modules coming from differential geometry

Let $V$ be a $n$-dimensional real vector space equipped with a positively definite scalar product $g$ and let $\mathrm{O}(n)$ be the automorphism group of $(V,g)$. View $V^{\otimes k}$ as a ...
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2answers
363 views

Chinese Remainder Theorem backwards

I have the following situation, that is much alike the Chinese Remainder Theorem. Let $\phi_d(\alpha)$ be the $d^{th}$ cyclotomic polynomial in the variable $\alpha$ (I'm not specifying the ...
5
votes
1answer
174 views

Quadratic Casimir of fundamental irreps of simply-laced Lie algebras

I have the following question, motivated by the expression for the character of level 1 highest weight integrable representations of simply-laced affine algebras (in terms of the string function). It ...
5
votes
0answers
113 views

Generators for invariant tensors of lie algebras

EDITED FOR (hopeful) CLARITY: For a simple Lie algebra $\mathfrak{g}$ (over $\mathbb{C}$) and its adjoint group G, the $G$-invariant polynomials on $\mathfrak{g}$ are linear combinations of products ...
3
votes
1answer
165 views

Lie group GL(4) representation decomposition

Let $V$ be the defining representation of $GL(4,\mathbb C)\to GL(V)$ with $V=\mathbb{C}^4$. Let $Ext\;V$ be the exterior square of $V$ which is a 6-dim repsentation. My question: How does $$V\otimes ...
3
votes
1answer
64 views

Intersection of Maximal Left Ideals with Finite Dimensional Quotient

Let $\Gamma$ be a finitely generated group and let $A=\mathbb{C}[\Gamma]$ be the corresponding group algebra over $\mathbb{C}$. Let $X$ be the set of all maximal left ideals of $A$ and let $X_0=\{I ...
10
votes
1answer
241 views

Representation varieties of 3-manifold groups in SL(n,C)

I am looking at the variety of representations of the fundamental group of a hyperbolic 3-manifold into $SL(n,C)$: $$Hom(\pi_1M, SL(n,{\mathbb C}))$$ It is known that volume and Chern-Simons ...
2
votes
1answer
99 views

Invariants of polynomial algebras on infinite G-modules

Is there a reasonable characterization of the ring of $G$-invariants ($G$ - a finite dimensional simple Lie group) in the polynomial algebra on direct sum of infinite number of irreducible ...
3
votes
1answer
203 views

Is there a left-adjoint to the restriction of comodules?

If $f\colon C \to D$ is a homomorphism of coalgebras and $\rho\colon V \to V\otimes C$ is a $C$-comodule, then $(1\otimes f)\rho\colon V \to V\otimes D$ is the comodule restricted to $D$. In e.g. ...
2
votes
1answer
149 views

Computing tangent spaces of resolutions to Slodowy slices

This question is about (a special case of) the varieties discussed here: Does the preimage of the Slodowy slice in $T^*G/P$ have a name?. Let $G = SL_n(\mathbb{C}), \mathfrak{g} = ...
4
votes
2answers
202 views

What is the map from nodes of the E8 diagram to conjugacy classes in the binary icosahedral group?

Let $G \subset \mathrm{SL}_2(\mathbf{C}^2)$ be a finite subgroup isomorphic to the binary icosahedral group. Let $Y$ be the minimal resolution of $\mathbf{C}^2/G$. The irreducible components of the ...
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0answers
67 views

Contragredient via an involution?

So in reading about epsilon factors of pairs for $G=GL_n(F)$ over a local nonarchimedian field $F$, the following fact was used. If $\pi$ is an irreducible smooth (complex) $G$-representation, then ...
1
vote
1answer
121 views

classification of irreducible finite dimensional representation of affine hecke algebra of type A

Let $H_{n}$ be the affine Hecke algebra with parameter q, where q is not root of unity. The classification of irreducible finite dimensional representations has been given by Kazhdan-Lusztig in terms ...
2
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4answers
641 views

Are there natural examples of non-symmetric Frobenius algebras?

Symmetric Frobenius algebras arise everywhere, but the non-symmetric variety seem difficult to come by. Are there any natural examples/constructions that produce non-symmetric Frobenius algebras in ...
6
votes
1answer
119 views

About G-modules with good filtrations

Let $k$ be an algebraically closed field of positive characteristic, and let $G$ be a reductive algebraic group over $k$ (for instance a classical group). Let $V$ be a (rational) $G$-module. We say ...
9
votes
1answer
159 views

Fixed set of order p automorphism of Bruhat-Tits tree

I would like to know the structure of the fixed set of an order $p$ automorphism [Edit: induced by a matrix in $GL_2(K)$] on the Bruhat-Tits tree for a p-adic field $K$, specifically in the case where ...
3
votes
2answers
235 views

Algebraic Groups, Modules, and Comodules

Background: Let $H$ be a finitely generated commutative Hopf $k$-algebra, where $k$ is a field of non-zero characteristic. For $$ \widehat{H} := \text{Alg}_k\{H; k\}, $$ we recall (see Abe Chapter 4 ...
2
votes
1answer
172 views

R-linear representations of sl(2,C)

Is there some good reference for the classification of finite-dimensional ${\mathbb R}$-linear (as opposed to ${\mathbb C}$-linear) representations of $\mathfrak{sl}_2{\mathbb C}$? Equivalently, what ...
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vote
2answers
245 views

A semisimple group ring

Let $n \in \mathbb{N}$, $p$ a prime number, and $G$ a finite group of order coprime to $p$. Let $R = \mathbb{Z} /p^n \mathbb{Z}$ be the ring of integers mod $p^n$. Must $R[G]$ be semisimple? As noted ...
1
vote
1answer
133 views

A Simple Bijective Proof Of Stanley's Hook-Content Formula for Hook Shapes

this is my first post on math overflow so I hope it goes well. I believe I have a fairly simple bijective proof for Stanley's Hook-Content Formula in the case of hook shapes. I wanted to see if ...
10
votes
2answers
253 views

Which finite nonabelian groups have all their quaternionic representations of degree one?

A finite group $G$ has a finite set of irreducible representations over the complex numbers. All of these representations are linear (that is, are maps in 1x1 complex matrices) if and only if $G$ is ...
0
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0answers
90 views

Actions and representations of profinite groups

Let $p$ be a prime number, and denote by $\mathbb{Z}_p$ the additive profinite group of p-adic integers. Let $G$ be a finitely generated profinite group of order coprime to $p$, and $V = ...
3
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0answers
352 views

An exact sequence which does not split

Let $X$ and $Y$ be indecomposable modules over a finite dimensional algebra and let $f \colon X \to Y$ be a non-zero morphism which is neither a monomorphism nor an epimorphism. Suppose that it is ...
3
votes
1answer
150 views

Integral representations of groups of small order

I have a problem in which it would be helpful to know about the integral representations of some groups of small order (probably of fairly low degree). From what I've gathered so far, cyclic groups of ...
2
votes
1answer
221 views

Quiver representations

I'm wondering how to find indecomposable representations of a given quiver explicitely. In particular, I'm interested in the maximal indecomposable representation of $\mathbb{E}_8$(I'm working over ...
4
votes
3answers
161 views

Questions on constructions of supercuspidal representations

To my knowledge, usually there are two ways to construct supercuspidal representations over p-adic fields. The first is via theory of types (for GL(n) and classical groups), notably by Bushnell, ...
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207 views

Relationship between algebraic groups and Lie groups? [closed]

In the literature, e.g. in representation theory, there seems to be a passage from Lie groups to (linear) algebraic groups. It is clear, particularly over $\mathbb R$ and $\mathbb C$ that they are ...
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134 views

A decomposition of a representation via characters of a normal compact subgroup

This is connected to my question here. Let $K$ be a normal compact subgroup in a locally compact group $G$, $\widehat{K}$ the dual object for $K$, and $\mu_K$ the normed Haar measure on $K$ ...
6
votes
1answer
258 views

What is the Grothendieck group of the category of $\mathbf{Z}_p[G]$-modules?

Let $G$ be a finite group. Let $\mathcal{O}$ be a suitably large finite extension of the $p$-adic integers, with residue field $\mathbf{F}_q$. The Grothendieck group of the category of ...
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74 views

“Multiplying” Clebsch-Gordan series

Assume you have a Lie algebra $G$ and a Clebsch-Gordan series $A\bigotimes{B}=C\bigoplus{D}\bigoplus...$ Assume you have a Lie algebra $g$ and a Clebsch-Gordan series ...
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435 views

When is the conjugation character almost multiplicity free?

Let me give some motivation, which also explains how I arrived at the question. We may let the finite group $G$ act on itself by conjugation, and this makes the group ring into a $\mathbb{Z}G$-module ...
5
votes
1answer
285 views

Is there a specific geometric meaning why fractional charges are allowed in SU(N) gauge theories?

So in the standard model of particle physics, there exist particles with fractional charge. What this means geometrically is as follows: We are given a smooth manifold with a principal $U(1)$ bundle ...
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2answers
181 views

Minimal *-idempotents for the group algebra of the symmetric group

There is a well-known construction of minimal idempotents in the group algebra of the symmetric group $\mathbb C[S_n]$ using row symmetrizers and column antisymmetrizers. But these idempotents are ...
4
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1answer
167 views

Condition on a Hopf operad for tensor product in the base categoy to be a (categorical) coproduct for algebras

A Hopf operad will be an operad endowed with a coproduct $P(n) \longrightarrow P(n) \otimes P(n)$ which is compatible in the obvious sens with operad laws (no more structure is assumed a priori. ...