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        Questions tagged [rt.representation-theory]

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

        3
        votes
        0answers
        65 views

        The arithmetic meaning of opers (if any)

        Let $G$ be a complex, connected semi-simple Lie group, $G'$ its Langlands dual group, $\mathrm{Bun}_G$ the moduli stack of $G$-bundles on a smooth projective curve $\Sigma$ over complex numbers, $\...
        1
        vote
        0answers
        72 views

        When is the Jacquet module semisimple?

        Let $G$ be a reductive algebraic group over a $p$-adic field, $N$ be the unipotent radical of a parabolic subgroup $P$ of $G$, $M$ be a Levi subgroup of $P$ so $P = M \rtimes N$, and $\chi$ be a ...
        3
        votes
        0answers
        28 views

        Cartan determinants of minimal Auslander-Gorenstein algebras

        Iyama and Solberg introduced minimal Auslander-Gorenstein algebras as algebras having finite dominant dimension ($\geq 2$) equal to the Goreinstein dimension in https://www.sciencedirect.com/science/...
        6
        votes
        1answer
        99 views

        Relative Dickson (trace) criterion for Jacobson radical?

        In the following, all algebras are associative and unital. Let $J\left(A\right)$ denote the Jacobson radical of an arbitrary algebra $A$. Recall that this is defined as the set of all $a \in A$ such ...
        3
        votes
        1answer
        63 views

        Distance between Verma modules in certain “strongly” standard filtrations

        On p. 128 of the book: Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$. I quote: "......Delorme arrives at vanishing criteria for Ext$^n(\mathcal{O})$ which are more ...
        2
        votes
        0answers
        24 views

        Extenstion functor on quotient map

        Let $\rho$ be the half sum of positive roots in $\Phi^+$, $M_x$ be the Verma module with highest weight $x\cdot(-2\rho)$ and $L_w$ be the simple highest weight module with highest weight $w\cdot(-2\...
        4
        votes
        0answers
        54 views

        Stable equivalence and stable Auslander algebras

        Let $A$ be a representation-finite finite dimensional quiver algebra and $M$ the basic direct sum of all indecomposable $A$-modules. Recall that the Auslander algebra of $A$ is $End_A(M)$ and the ...
        4
        votes
        0answers
        54 views

        Derived equivalence between two exotic algebras

        Let $A$ and $B$ be two connected finite dimensional quiver algebras having the same underlying quiver. Question 1: In case $A$ and $B$ have exactly one indecomposable projective non-injective $A$-...
        3
        votes
        0answers
        158 views

        Kazhdan–Lusztig polynomials in terms of Ext groups

        Let $P_{x,w}$ be the Kazhdan–Lusztig polynomial, $\rho$ be the half sum of positive roots in $\Phi^+$, $M_x$ be the Verma module with highest weight $x\cdot(-2\rho)$ and $L_w$ be the simple highest ...
        4
        votes
        1answer
        109 views

        Typical and atypical modules for Lie superalgebras

        There are two types highest weight representations for a Basic classical simple Lie superalgebra $\mathfrak{g}$ which are defined as typical (representation for which highest weight vector is the only ...
        2
        votes
        2answers
        72 views

        Symmetry of Casimirs of Lie algebras

        The dimensions of the invariant tensors (Casimirs) of the simple Lie algebras are known, but I nowhere could find whether they are completely symmetric or antisymmetric with respect to an variable ...
        1
        vote
        0answers
        83 views

        Hilbert modular form as a representation of Hecke algebra

        I am reading some notes by Snowden and I don't understand a sentence. Clearly, if we have an appropriate $R = T$ theorem then we get a modularity lifting theorem, as $\rho$ defines a homomorphism ...
        4
        votes
        1answer
        166 views

        Do we have $G(\mathbb A_S) G(k) = G(\mathbb A)$ for sufficiently large $S$?

        Let $G$ be a linear algebraic group over a number field $k$. If necessary, assume $G$ is connected and reductive. Let $\mathbb A$ be the ring of adeles of $k$, and $\mathbb A_S = \prod\limits_{v \in ...
        2
        votes
        1answer
        56 views

        Translation functor on parabolic Verma module

        I want to prove that following proposition by using Theorems/propositions in Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$. Define $ \Lambda := \{\nu \in \mathfrak{h}^* ...
        4
        votes
        0answers
        43 views

        Intersection of components in Springer fibre of type A

        From the standard results on Springer fibers of type A, we know that given a Springer fiber, say $\mathcal{B}_\lambda,$ its irreducible components are all equidimensional and parametrized by standard ...

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