# Tagged Questions

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

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### Cominuscule property of nilpotent orbits

Let $G$ be a complex reductive Lie group, $G/P$ a flag manifold, and $\Phi: T^* G/P \to {\mathfrak g}^*$ the moment map. So $\Phi(T^* G/P)$ is the closure of a nilpotent orbit. Lots of classes of ...
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### Set of Special Unitary Matrices that are dense in SU(4) and obey certain relations

I'm trying to find a finite set of 4x4 Unitary matricies $\{U_1,U_2,\ldots U_N\}$ such that the matrices are dense in SU(4), and obey the relations: $[U_i, U_j] = 0$ for $|i-j|>1$ ...
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### Reference request for an introduction to deformation theory in algebraic geometry

I'd like some introductory references for deformation theory in algebraic geometry. I'm interested in survey articles too but I primarily want references which give all the definitions and go through ...
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### Is there a finite test for isomorphisms of rigid monoidal abelian categories?

Let $G$ be a semisimple algebraic group. (I'm already interested in the case $G=SL_2$.) Let $\mathcal C$ be a semisimple rigid monoidal abelian category endowed with pair of exact tensor functors ...
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### Nice proofs of the Poincaré–Birkhoff–Witt theorem

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra with an ordered basis $x_1 < x_2 < ... < x_n$. We define the universal enveloping algebra $U(\mathfrak{g})$ of $\mathfrak{g}$ to be the ...
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### Most discriminants are almost squarefree

Write, for $f(x) = x^d + a_2 x^{d-2} + \cdots + a_d\in \mathbb{Z}[x]$, $H(f) := \max(|a_i|^{\frac{1}{i}})$. Does anyone know of a reference that would allow me to show that the proportion of $f$ with ...
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### Invariants and stablizers for the $PGL(V)$ action on $End(V\otimes V)$

Let $K$ be a field of characteristic zero, and $V$ a finite dimensional vector space over $K$. Consider the action of the algebraic group $G:=PGL(V)$ on the vector space $W:=End_K(V^{\otimes 2})$ by ...
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### Birch's conjecture from Representation Theory

Birch has a conjecture about which automorphic forms on $PGL(2)$ are the lifts from nonsplit $O(3)$. Temporarily ignore global issues, and focus on the local nonarchimedian picture. The automorphic ...
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### Conjugacy scheme, fppf versus GIT

I would be glad to have some guidance in the following. Let $k$ be an algebraically closed field. Let $G$ be a connected reductive group over $k$. Denote by $\mathfrak{c}$ the Zariski spectrum of the ...
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### Construct discrete series of SL(2,R) as kernel of twisted Dirac operators

I’m studying the paper of (Baum-Connes-Higson, ex 4.25), and I would like to give an explicit computation for the Connes-Kasparov conjecture for SL(2,R). The idea is that each non-trivial ...
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### Technical issue in the approach to Lie groups taken in Brian C. Hall's book

I'm teaching Lie groups and Lie Algebras out of Brian C. Hall's book, which I've enjoyed using. I'm confused about a technical hitch though that I'm not sure how to avoid. The approach taken in this ...
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### Are the distributive permutation groups linearly primitive?

An action of a group $G$ on a set $X \neq \emptyset$ is called transitive if $\forall x,y \in X$, $\exists g \in G$ such that $g.x = y$. It is called primitive if it is transitive and preserves no ...
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### How do I determine a real matrix form for a group representation?

Hello mathoverflow community, I am a little stucked working on my master thesis. For a representation on $\mathbb{Z}_p\ltimes\mathbb{Z}_p^*$ induced from the additive character $\chi$ of ...
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### Bott-Samelson construction of a perfect Morse function on G/T

An undergraduate student of mine is interested in writing a "senior" thesis on the topology of Lie groups. Let $G$ be a simply connected compact Lie group and $T$ a maximal torus. One way of ...
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### Invariant ring of $S_5$

The irreducible representations of the Symmetric group $S_5$ are classified by the partitions of $5$. For the standard representation which corresponds to the partition (4,1) the ring of invariants is ...
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### A Zero-Multiplicity Problem Related to Foulkes' Conjecture

I'm a combinatorialist that is interested in estimating multiplicities of irreps of $1^{S_{kn}}_{S_k \wr S_n}$ (the action of symmetric group on uniform partitions). I'm aware of the difficulty (or ...
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Physicists routinely wrote all 3 Pauli spin matrices as a vector. $$\sigma_1 = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0\end{array} \right) \hspace{0.25in} \sigma_2 = \left( ... 0answers 42 views ### Extra-Lorentzian Kac-Moody algebras My question is about Kac-Moody (KM) algebras of finite rank with symmetrized Cartan matrices B = C A (A is Cartan matrix) of signatures (-,-,+,...,+), (-,-,-,+,...,+), etc. i.e. with ... 0answers 69 views ### the linear span of all matrix coefficients is C(G,\mathbb{C}) where G is a finite group Theorem. Let \{(R_{\alpha},V_{\alpha})\} be a complete set of inequivalent irreducible finite dimensional representations of a finite group G. Let V_{R_{\alpha}} be the subspace generated by all ... 0answers 51 views ### Computing Springer action on the homology of affine Springer fibers Lusztig defined (in Sec. 5, also Sage) a Springer action of the affine Weyl group on the homology of affine Springer fibers (Iwahori one, i.e. in an affine flag variety). In the regular semisimple ... 1answer 405 views ### Plugging 1-x into Schur polynomials I have a symmetric Laurent polynomial f in k variables expressed as a linear combination of Schur polynomials. I'd like to know what happens when I make the substitution p(x_1,\ldots,x_k)\mapsto ... 2answers 228 views ### Fundamental invariants for root subsystems Let \Phi be an irreducible root system of rank \ell. The fundamental invariants of \Phi is a set of \ell integers d_1, \cdots, d_\ell canonically attached to \Phi. Now suppose \Psi is ... 0answers 123 views ### An analogue of Deligne--Lusztig theory for real groups? I am considering the following analogue of Deligne--Lusztig theory: Take e.g. G=GL_n(\mathbb{C}), and let F be the complex conjugate, then we have G^F=GL_n(\mathbb{R}). Consider the Lang ... 1answer 158 views ### Wavefront sets of irreducible representations with non-integral infinitesimal characters Let G be a complex reductive algebraic group (connected, simply connected etc), viewed as a real group. We study the representations of G, and we follow the notations in the paper of Barbasch and ... 1answer 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, ... 0answers 45 views ### How to compute t_0 and r^0 in Belavin-Drinfeld's classification of solutions of classical Yang-Baxter equations? I tried to understand Belavin-Drinfeld's classification of solutions of classical Yang-Baxter equations. In the book a guide to quantum groups, on page 83, there is an example of solutions of the ... 1answer 116 views ### Must normalizing field outer automorphisms “divide” the dimension? Imprecise question: To get a normalizing field outer automorphism of order r, must we multiply the dimension by r? Precise hypothesis: Let p\geqslant 5 be a prime, let q be a power of p and ... 1answer 81 views ### Weingarten function for unitary group Studying integration over unitary group I came across this function, the Weingarten function Wg, such that$$ \int_{\mathcal{U}(N)} \prod_{k=1}^{n} U_{i_kj_k} U^*_{m_k r_k} dU=\sum_{\tau,\sigma\in ...
Let $\mathbb{F}$ be a field. The Tits building for $\text{SL}_n(\mathbb{F})$, denoted $T_n(\mathbb{F})$, is the simplicial complex whose $k$-simplices are flags 0 \subsetneq V_0 \subsetneq \cdots ...