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        Questions tagged [rt.representation-theory]

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

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        votes
        2answers
        41 views

        Decomposition into irreducible of a representation of the wreath product $S_d\wr S_n$

        Let $S_d, S_n$ be the permutation groups of $d,n$ elements. An intuitive representation of the wreath product $S_d\wr S_n$ is $V_1\otimes...\otimes V_n$, where each $V_i$ is of dimension $d$. Writing ...
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        votes
        2answers
        544 views

        Units in group rings.

        Let $G$ be a finite solvable group of order $n$, and let $g_1 ... g_n$ be an enumeration of its elements. Let $a_1 ... a_n$ be a sequence of integers, such that $\sum a_i$ is relatively prime to $n$. ...
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        vote
        0answers
        51 views

        Do we have $K \cap P = (K \cap M)(K \cap N)$?

        Let $G$ be a connected, reductive group over a $p$-adic field $k$, let $P$ be a parabolic subgroup with Levi $M$ and radical $N$. Let $K$ be a maximal open compact subgroup of $G$ in good position ...
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        votes
        0answers
        64 views

        Global dimension of the tensor algebra

        Let $R$ be a semisimple ring with a non-zero $R$-bimodule V. Let $T_R(V):= \bigoplus\limits_{k=0}^{\infty}{V^{\otimes_k}}$ be the tensor algebra of $V$. Question 1: Is there a simple proof that $...
        3
        votes
        1answer
        102 views

        Global dimension of a graded algebra

        Let $A= \bigoplus\limits_{n=0}^{\infty}{A_n}$ be an $\mathbb{N}$-graded algebra with semisimple $A_0$. Question: Do we have that the global dimension of $A$ is equal to $\sup \{i \geq 0 | Ext_A^i(...
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        votes
        1answer
        123 views

        Finite subgroups of $PSU(3)$

        I'm looking for a reference to a classification or description of finite subgroups of $SU(3)$ that contain the center, or equivalently $PSU(3)$. Can anyone point me in the right direction?
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        votes
        0answers
        133 views

        The $\frak{sl}_2$-representation on a symplectic manifold

        Any symplectic manifold $(M,\omega)$ carries a representation of $\frak{sl}_2$: Define the maps $$ L: \Omega^\bullet \to \Omega^{\bullet}, ~~~~~~~~~ \Lambda: \Omega^\bullet \to \Omega^{\bullet}, ~~~~~~...
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        votes
        1answer
        184 views

        Quadratic algebras and Koszul algebras

        Let $A$ be a quadratic algebra and $B$ the Ext-algebra of $A$. In case $A$ is a Koszul algebra, we should have that the global dimension of $A$ plus one is equal to the Loewy length of $B$ (is there a ...
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        votes
        2answers
        68 views

        Definition of the weight lattice for a nonreduced root system

        Let $(V,\Phi)$ be a root system with dual root system $(V^{\ast},\Phi^{\vee})$. Let $\Delta = \{\alpha_1, ... , \alpha_n\}$ be a set of simple roots for $V$, and let $\Delta^{\vee} = \{\alpha_1^{\vee}...
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        votes
        0answers
        24 views

        About Extension group and weights in $\mathcal{O}^\mathfrak{p}$

        Denote $M_I(\lambda)$ be the generalized Verma module with highest weight $\lambda$ and $L(\mu)$ is the simple highest weight module with highest weight $\mu$. Suppose $\text{Ext}_{\mathcal{O}^\...
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        vote
        0answers
        48 views

        Finite-dimensional graded Lie algebras with $2$ generators

        Does anyone know of a classification of those (complex) Lie algebras which are: generated by two elements $\mathbb{Z}$-graded Lie algebras finite dimensional
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        votes
        1answer
        96 views

        P-adic representations corresponding to the same cuspidal pair

        Let $G(F)$ be a reductive $p$-adic group. A result of Bernstein says that we can correspond each smooth irreducible representation to a “cuspidal pair” where it is embedded, and at most finitely many ...
        6
        votes
        1answer
        134 views

        Irreducibility of Gelfand-Serganova strata

        To keep the notations simple I'll restrict my attention to the complete flag variety although the question should be equally valid for partial flag varieties. Consider $G=SL_n(\mathbb C)$ with Borel $...
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        0answers
        27 views

        Rigid $Hom$-orthogonal modules in wild hereditary algebras

        Let $Q$ be a simply-laced wild quiver with at least one multiple edge, $k$ be an algebraically closed field, $1$ be the source of and $2$ be the sink of one set of such edges. Can we find rigid ...
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        votes
        0answers
        40 views

        Properties of extendable irreducible characters to a normal Sylow subgroup

        Let $G$ be a finite group with a nilpotent commutator subgroup, and denote by $mcd(G)$ the set of degrees of the irreducible monomial characters. Suppose $|mcd(g)|=2$. Furthermore, we call a monomial ...

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