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

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9 views

equivalence of Lie group and Lie algebra intertwiner [migrated]

I encountered this problem while working on my research. Let $G$ be a Lie group, and consider an intertwiner of the complex representations (possibly infinite-dimensional) $$ \pi:G\rightarrow ...
0
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0answers
29 views

Relating the R-Transform in Free Probability to noncommutative group representations

In traditional (commutative) probability theory, sums of random variables correspond to convolution of distribution functions, which plays well with the Fourier Transform. In free (noncommutative) ...
2
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1answer
124 views

Is every weight of an integrable highest weight module in the Tits cone?

Let $\mathfrak{g}$ be a Kac-Moody algebra with Cartan subalgebra $\mathfrak{h}$, Weyl group $W$, and simple roots and coroots $\alpha_i, \check{\alpha_i}, i \in I$, respectively. Let $L$ be an ...
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0answers
33 views

Largest Set of Special Unitary Matricies With Invariant Subspace For Adjoint Action

I am trying to solve the following. Given the special unitary group $SU(n)$ and its adjoint action $Ad_{U}: \mathfrak{su}(n) \rightarrow \mathfrak{su}(n)$, what is the largest subset of $SU(n)$ such ...
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1answer
259 views

On a claim of Zagier on extending a map to cocycle

Zagier, in his paper 'Some Surprising Consequences of the Cohomology of SL$_2(\bf{ Z})$' (link, p. 6), studies the action of $\Gamma=PSL_2(\bf Z)$ on a vector space $V$, denoting the action by $v\ |\ ...
2
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2answers
154 views

Permutation covering of a $G$-lattice

Let $G$ be a finite group. By a $G$-lattice we mean a finitely generated free abelian group $L$ with an action of $G$. We say that $L$ is a permutation $G$-lattice if $L$ has a ${{\mathbf{Z}}}$-basis ...
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0answers
43 views

Distiguishing mutant knots

Can an invariant from a quantum Lie algebra ever distinguish mutant knots? (Maybe if it is "chiral"...whatever that means :-) (Note that Kauffman abstract tensors/skein equations CAN distinguish ...
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0answers
74 views

a question about Nakayama functor [closed]

Assume $A$ a finite dimesional algebra over $k$, we assume $k$ algebraically closed, then can we computer $Hom_k(RHom_A(Hom_k(A,k),A),k)$ which is $N^2(A)$, here N is the derived Nakayama functor. For ...
3
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0answers
155 views

Can the product of a simple and a non-simple indecomposable representation be semisimple?

Consider two (possibly infinite-dimensional) representations $\rho$, $\pi$ of a semisimple Lie algebra $\mathfrak{g}$, with $\rho$ irreducible and $\pi$ indecomposable but not irreducible (i.e., not ...
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0answers
97 views

Fundamental theorems of invariant theory, twisted by a representation

Let $V$ be a finite dimensional complex vector space and let $G = \mathrm{GL}(V)$. The first and second fundamental theorems of invariant theory for $V$ give generators and relations for the algebra ...
2
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1answer
140 views

Understanding the Weyl Character Formula

Let $G$ be a compact (connected) Lie group with a maximal torus $T$. For each (analytically) integral weight $\lambda$ the Weyl character formula $$\Theta_{\lambda}(H)=\frac{\sum_{w\in ...
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2answers
218 views

Representations of complex semi-simple algebraic group “defined over $\mathbf{Z}$”?

If $G$ is a split semisimple linear algebraic group over $\mathrm{Spec}(\mathbf{Z})$ then does every (algebraic) irrep of $G_{\mathbf{C}}$ extend to a morphism $G\to\mathrm{GL}_n$ over ...
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2answers
214 views

Representation Theory of $U(N)$

(1) Is it true that the category of representations of $U(n)$ is equivalent to the category of representations of $SU(N) \times U(1)$? If so, how is it proved, or what is a good reference. (I guess ...
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1answer
60 views

Using Magma to Find a Fixed Points Module [closed]

Let $G$ be a group and $H$ a subgroup. Suppose $M$ is a $kN_G(H)$-module ($k$ a field). Then the $H$-fixed points in $M$ denoted $M^H$ is a $kN_G(H)$-module. Is there a way to access this module in ...
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0answers
43 views

Kac-Moody D6 level 2 - “copying” objects necessary for some based rings to categorify?

This is the relevant data for my example (but the question applies more general). So you have the MTC $D_6$ at $q=(-1)^{1/12}$ (my q) and there are listed 13 objects (1 trivial, 3 dim=1, 4 ...
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73 views

Approximating the norm of a finite dimensional representation on a Banach space by irreducible representations

Let $G$ be a compact group, let $X$ be a Banach space and let $\pi$ be a linear and isometric representation of $G$ on $X$ that is continuous with respect to the strong operator norm. For $v \in X$, ...
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0answers
79 views

What is the definition of plethysm in the representation theory of permutation groups

Let $s_\lambda \circ s_\mu$ be a plethysm. Here let $\lambda, \mu$ be $m,n$ box Young diagrams. I have seen the definition of plethysms in symmetric functions. I would like to understand the ...
2
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0answers
35 views

Noncommutative fusion categories

Although noncommutativeness is almost a defining trait for fusion categories, offhand I recall only the extended Haagerup N-N (rank 8). It's a two minute computation to find that even a based ring ...
1
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1answer
107 views

Clifford-Mackey theory, references

I am working on a problem related to the local Langlands correspondence and I am interested in certain smooth representations of locally profinite groups (in particular of the Weil group of a local ...
1
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1answer
123 views

Ext groups in the equivariant derived category

I apologize in advance that this question is probably too basic for MO, but I reckoned I would not get an answer on Math.Stackexchange. I am starting to learn about perverse sheaves, the ...
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0answers
44 views

Non-graded representations over Lie superalgebra $\mathfrak{gl}(m,n)$

I have the following questions: Let $m,n$ be positive integers. Consider representations over the general linear Lie super-algebra $\mathfrak{gl}(m,n)$. Namely, modules over the associative algebra ...
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0answers
90 views

Fusion: Hexagon implies pentagon?

Or in decent language: :-) Does the Racah backcoupling rule(s) imply Biedenharn-Elliot? Or for based rings: Can a based ring be braided and still not be a fusion category? (The 2-1-4-8 here would be a ...
27
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4answers
855 views

Why there is a relation among the second-order minors of a symmetric $4\times 4$ matrix?

A $4\times 4$ symmetric matrix $$ \left( \begin{array}{cccc} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{12} & a_{22} & a_{23} & a_{24} \\ a_{13} & a_{23} & a_{33} & ...
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121 views

An equivalence of derived categories by Happel-Reiten-Smalø

I have a problem in understanding the proof of a theorem by Happel-Reiten-Smalø. The original reference is this article http://arxiv.org/abs/0911.4473 . I write down the text of the theorem and a ...
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0answers
101 views

Quantum Half :-)

"Quantum Half" of course is the reverse of "quantum double". 1. Any quantum double is braided, but I guess the reverse is very untrue - if you have a braided fusion ring F, only rarely there exists a ...
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0answers
108 views

Finite quotients of an infinite product of finite groups

Let $G$ be a finite group. Consider the direct product $\Gamma=\prod_{i=1}^{\infty}G$ of (countably) infinitely many copies of $G$. For every finite set of numbers $\{i_1,\ldots,i_n\}$ we have the ...
3
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0answers
125 views

The special embedding $\mathfrak{so}(7)\subset\mathfrak{so}(8)$

It is commonly known that we have a chain of embeddings $$SU(4)\subset Spin(7)\subset SO(8)$$ (there is more than one possible $Spin(7)$, just take one). Which is the explicit analog for the Lie ...
2
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2answers
314 views

Lifting projective Galois representation with condition

Let $\bar{\rho}: G_K\to PGL_n(\mathbb{C})$ be projective representation of the absolute Galois group of a number field $K$ and $\varphi\in Aut(G_K)$. A theorem of Tate tells us that we can always ...
3
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1answer
148 views

Fusion categories: If infinity were an integer

Consider the following fusion categorie $F(i)$ with integer parameter $i$. Simple objects are $1,a,A,B$ (where $a$ and $A$ are conjugates). Nontrivial fusion rules are $a\bigotimes{a}=A$ (and ...
9
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1answer
301 views

Embedding linear algebraic groups of a given dimension into a fixed $\mathrm{GL}_N$

Given $n$, can $n$-dimensional linear algebraic groups over $\mathbb{C}$ be embedded into $\mathrm{GL}(N,\mathbb{C})$ for a uniformly bounded $N$? Thanks so much for your reply!
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1answer
252 views

Irreducibility of the tensor product of two finite-dimensional irreducible group representations

Let $k$ be an algebraically closed field of characteristic 0, let $G$ be any group and $N\unlhd G$ a normal subgroup. Let $U$ be a finite-dimensional and irreducible $kG$-module, such that $U$ is also ...
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3answers
284 views

Real and Quaternionic Representations according to Weights

According to this question, it is easy to know whether a representation is self dual or not: just check if the weight distribution in space is symmetric about the origin. Now, for self dual ...
4
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1answer
178 views

Do discrete groups with property $(T)$ have “modest” subgroup growth?

I saw it conjectured at http://www.mathunion.org/ICM/ICM1994.1/Main/icm1994.1.0309.0317.ocr.pdf that "discrete subgroups with property $(T)$ may have modest subgroup growth." (Page 5, directly above ...
2
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1answer
191 views

A computation about Whittaker functions and Eisenstein series

I have some questions about the computation of Eisenstein series and Whittaker functions in the book. The question is on page 29, Theorem 4.3. My questions are in the following. (1) I think that ...
4
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0answers
82 views

Uniqueness of cohomological holomorphic discrete series representation

In Claus Sorenson's PhD thesis, he proves a theorem about level lifting of paramodular forms whose associated automorphic representation has component $\pi_{\infty}$ that is the "cohomological ...
4
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0answers
101 views

Root-theoretic formulation of characteristic polynomial

Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra of rank $n$ over $\mathbb{C}$. Let $G$ denote the corresponding simple simply connected algebraic group. By Chevalley's Theorem, ...
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78 views

Polynomials invariant with respect to a nilpotent Lie algebra

Let $\mathfrak{u}$ be a nilpotent Lie algebra and let $\mathbb{C}[\mathfrak{u}]$ be the space of polynomials with the natural coadjoint action of $\mathfrak{u}$. Can one describe ...
2
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0answers
93 views

Classification of symplectic representations of quaternion division algebras

I would like to know the classification of representations of the form $\rho:B^{\times}\to Sp(V,F)$ or ($Gsp(V)$), where $B$ is a quaternion division algebra over a number field $F$ (or ...
4
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0answers
55 views

Minimal rank of a permutation resolution of a $G$-lattice

Let $G$ be a finite group. By a $G$-lattice I mean a finitely generated free abelian group $L$ with an action of $G$. One says that $L$ is a permutation lattice if $L$ has a $\mathbb{Z}$-basis ...
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0answers
38 views

General quantum highest-weights dimension formulas

The formulas hold modulo typos :-) It is well known (tl;dr fun fact: not well enough for me, I forgot where I saw it so I guess-computed it from the data in the Hayashi paper; promptly I found it in ...
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0answers
17 views

Decomposition of axial vector and vector representions of C$_{4v}$ group

Let $R$ be the orthogonal matrix corresponding to an operation in $O(3)$. If R is a proper rotation, then both vectors $\vec{V}$ and axial vectors $\vec{A}$ are transformed in the same way, $$ ...
2
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1answer
148 views

The lower bound of a group with characters of special degrees

Is there any lower bound for the order of a group with an irreducible character of degree $p$, where $p$ is a prime. Is there any similar result for $p^2$ or $p^3$ instead of $p$? Thanks for your ...
2
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2answers
268 views

Algebraic groups “generated” by a Lie algebra element

Here is a definition which I invented and which I would like to understand better. Let $ A $ be a complex affine algebraic group. Let $ X \in \mathfrak g $ be an element in its Lie algebra. We say ...
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36 views

Pseudo-braided fusion categories

A few definitions first, please replace with the standard terminology (and correct me if I confuse all the by-names of fusion categories :-) I call a complex number $z$ pseudo-cyclotomic if $|z|=1$. I ...
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0answers
102 views

Two questions about Whittaker functions

I am watching the video: Modeling p-adic Whittaker functions, Part I. I have two questions about Whittaker functions in the video. From 33:00 to 37:00, it is said that after changing of variables, ...
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3answers
693 views

Catalan numbers as sums of squares of numbers in the rows of the Catalan triangle - is there a combinatorial explanation?

This question arose from an answer to my recent question How many traces are there on Temperley-Lieb, Fuss-Catalan, Iwahori-Hecke, Birman-Wenzl-Murakami-Kauffman, ... algebras? What I need from that ...
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0answers
47 views

Non-trivial summand in End(\rho)

Given a finite group representation $\rho:G\to GL_n(\mathbb C)$ one knows that the trivial representation $\mathbb 1$ is contained in $End(\rho)$. Let $\rho'$ be the other summand, i.e., $\rho'$ is ...
3
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0answers
101 views

n-homology of a Harish-Chandra module

Let $G$ be a connected real reductive Lie group and let $K$ be its maximal compact subgroup. Let $P=MAN$ a parabolic subgroup. Let $K_M^0=M^0\cap K$ be connected component of the maximal compact ...
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182 views

References for 'Theory of $p$-adic Galois Representations by Fontaine & Ouyang'

Presently I am reading the 'Theory of $p$-adic Galois Representations by Fontaine & Ouyang'. I am finding it difficult for eg. the initial sections on $l$-adic geometric representation of finite ...
3
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82 views

Localized at $p$ integral representations of finite elementary $p$-groups

Let $C_p$ be a cyclic group of prime order $p$. Let $F=C_p^n=C_p\times\dots\times C_p$ ($n$ times). I would like to to classify finite dimensional representations of $F$ over ${\mathbb{Z}}$. However, ...