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        Questions tagged [rt.representation-theory]

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

        4
        votes
        0answers
        74 views

        Two seemingly different definitions of a left cell

        This is a question about two seemingly different notions of a left cell in a finite Weyl group and why they are the same. My question arose from reading a paper of W. McGovern titled "Left cells and ...
        3
        votes
        0answers
        82 views

        Automorphy Factor from Vector Bundles on Compact Dual

        So I'm coming from an algebraic geometry perspective and I'm trying to carefully piece together the story of interpreting automorphic forms as sections of vector bundles on Shimura varieties. I think ...
        2
        votes
        1answer
        96 views

        Simple modules for direct sum of simple Lie algebras

        I think that the following statement is true, but I do not know how to prove it. Let $\mathfrak{g}_1$ and $\mathfrak{g}_2$ be two real simple Lie algebras. If $M$ is a (infinite dimensional) complex ...
        4
        votes
        2answers
        238 views

        About parabolic Kazhdan Lusztig polynomials

        There are two types of parabolic Kazhdan Lusztig polynomials, namely, of type -1: $P_{x,w}^{I,-1}$ and of type $q$: $P_{x,w}^{I,q}$. See Kazhdan–Lusztig and R-Polynomials, Young’s Lattice, and Dyck ...
        2
        votes
        1answer
        82 views

        A weak Schur's lemma for non-semisimple finite dimensional algebras

        Let $B \subseteq C$ be an inclusion of finite dimensional (associative) algebras over a field $k$. Assume that $C$ is a free $B$-module. Let $\bigoplus_i U_i$ be a decomposition of $B$ into ...
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        vote
        2answers
        92 views

        Openly available software to work with Demazure modules

        Does someone know of any sort of software openly available online which can be used to compute various characteristics of Demazure modules for semisimple Lie algebras? Specifically, I'm interested in ...
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        votes
        0answers
        82 views

        Extension of Verma modules

        The category $\mathcal{O}$ is the category of all finitely generated, locally $\mathfrak{b}$-finite and $\mathfrak{h}$-semisimple $\mathfrak{g}$-modules, where $\mathfrak{g}$ is a complex semisimple ...
        5
        votes
        1answer
        147 views

        Definition of a Dirac operator

        So it seems that a Dirac operator acting on spinors on $\psi=\psi(\mathfrak{su}(2),\mathbb{C}^2)$ can be written in this case simply as: $D=\sum_{i,j} E_{ij}\otimes e_{ji}$, where $E_{ij}$ are ...
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        101 views

        Proving that $\lambda\mapsto \chi^\lambda(C)/f^\lambda$ is a polynomial

        Let $\lambda$ be a partition of $n$ and $\chi^\lambda$ be the character of $S_n$ associated to it. Given any conjugacy class $C$, I want to prove that $$\lambda\mapsto \frac{\chi^\lambda(C)}{f^\lambda}...
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        votes
        0answers
        91 views

        Computing affine Springer fibers

        I'm having some trouble computing affine Springer fibers, even in simple cases. For example, consider the group $G=SL_2$ over $\mathbb{C}$ and let $\mathcal{K}=\mathbb{C}((z))$ and $\mathcal{O}=\...
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        votes
        0answers
        57 views

        Schur function on unit circles

        Define $T^d$ as following $$ T^d = \left\{(t_1,\cdots,t_d)\in\mathbb{C}^{d}\mid |t_i|= 1 \mbox{ for all } i\right\} $$ For any partition $\lambda\vdash n$,The Schur function is defined $$ \...
        2
        votes
        1answer
        67 views

        Verma modules in category $\mathcal{O}^\mathfrak{p}$

        Let $\mathfrak{g}$ be a complex semisimple Lie algebra and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$. Fix a Borel subalgebra $\mathfrak{b}$ containing $\mathfrak{h}$ and fix a ...
        2
        votes
        0answers
        55 views

        Levi subgroup of Siegel parabolic of GSpin

        I consider the group $G=\mathrm{GSpin(V)}$ as in this question. We have the so called Siegel parabolic $P$ (after fixing a cocharacter) and the associated Levi $M$ (these can also be obtained using ...
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        votes
        3answers
        409 views

        Borel's presentation for the cohomology of a Flag Variety

        If $G$ is a simple complex Lie group, $T\subset B\subset G$ is a choice of Borel and maximal torus, and $W$ is the Weyl group, then 1) $H^{*}(G/B,K)=K[T^{\vee}]/(K[T^\vee]^W_{+})$ and 2) $K[T^\vee]^...
        7
        votes
        1answer
        100 views

        Characters of orthogonal groups as symmetric functions

        This question was asked on MSE some time ago, here, but got no attention. The Schur functions are characters of irreps of the unitary group, $s_\lambda(U)=Tr(R_\lambda(U))$. They are symmetric ...

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