Questions tagged [rt.representation-theory]
Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
7,343 questions
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Zeros of polynomials associated to finite quivers
Let $Q$ be a finite connected quiver with path algebra $KQ$.
Let $C$ be the Cartan matrix of $KQ$ and $M:=C^{-1}+(C^{-1})^T$, the symmetric matrix correesponding to the symmetric Euler form of Q (up ...
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Decomposition of the Weil representation of Sp$(4, p)$ into irreducible representations of the SL$(2,p)$ subgroup acting on a 2d isotropic subspace
$\DeclareMathOperator\Sp{Sp}$In a previous question I asked (and was answered by LSpice) what is the decomposition of The Weil representation of $\Sp(4, p)$ into irreducible representations.
What I ...
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Block diagonalization of equivariant mapping of one isotypic component
Let $K$ be a field, $A$ a semisimple $K$-algebra, $p$ its minimal primitive idempotent, $M=Ap$ a minimal left ideal, and $D=End_A(M)$ a K-division algebra. Assume a $K$-vector space $V$ to be ...
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A linear algebra question related to the Bruhat factorisation of lower triangular matrices
First some abstract definition that generalise some notions from Rowmotion and Echelonmotion:
Definition 1:
Let $M$ be an $n \times n$ lower triangular integer matrix with entries in $\{0,1\}$ and ...
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Does off-diagonal trace randomness force a $t$-design?
Let $d\ge 2$ and let $\mathcal U=\{U_1,\dots,U_K\}\subset U(d)$ be a finite set of unitaries. For an integer $t\ge 1$ consider the $t$-th moment operator
$$
M_t := \frac{1}{K}\sum_{a=1}^K U_a^{\otimes ...
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Is the resolution of Kato's exotic nilpotent cone symplectic?
S. Kato introduced the exotic nilpotent cone and its resolution of singularities in his papers such as "An exotic Deligne-Langlands correspondence for symplectic groups".
My question is: is ...
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Half-integer Tate twist in the analytic geometric Satake equivalence
Let $G$ be a split reductive group over a (suitable) field $k$, the geometric Satakeequivalence of Mirkovic-Vilonen states that there is an equivalence of Tannakian categories
$$\operatorname{Perv}(\...
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How do I remove the topologically-nilpotent hypothesis from K-L's proof of equidimensionality of affine Springer fibers?
Let $G$ be simply connected semisimple over local field $F$. In Kazhdan--Lusztig's foundational paper on affine Springer fibers, they assume throughout that $\gamma \in \mathfrak{g}(F)$ is ...
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Submodule of the top chains of the Tits building
Let $F$ finite field, $G=GL_n(F)$, $B$ the subgroup upper triangular matrices in $G$, and $W$ be the subgroup of permutation matrices. Does $x = \sum_{w \in W} wB \in \mathbb C[G/B]$ always generate ...
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Decomposition of the Weil representation of $\mathrm{Sp}(4, p)$ into irreducible representations
What is the decomposition of the Weil representation of $\operatorname{Sp}(4, p)$ into irreducible representations?
The only things I (think I) know is that all the multiplicities involved are 1. ...
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Quiver and relations for the free idempotent monoid on n generators
Let $M_n$ be the free idempotent monoid on $n$ generators and let $A_n$ be the monoid algebra over the complex numbers.
$M_n$ is finite, see for example the answer in https://math.stackexchange.com/...
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Invariants of symplectic group
Let $k,n \in \mathbb{N}$. Consider the conjugation action of $\mathrm{Sp}_{2n}$ on $\mathrm{Sp}_{2n}^k$ and the corresponding invariant algebra $\mathbb{C}[\mathrm{Sp}_{2n}^k]^{\mathrm{Sp}_{2n}}$. Is ...
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Deformation of vertex algebra
The simple affine VOA associated to $\mathfrak{sl}(2)$ at level $1$ admits an adjoint action of the algebraic group $SL(2)$ [in fact $PSL(2)$]. The fixed point VOA is the universal Virasoro VOA at ...
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The centralizer $Z_{(n-1,1)} = Z[\mathbb C[S_n],\mathbb C[S_{n-1}]]$ as a torsion module over $Z\mathbb{C}[S_{n-1}]$
Let $ S_n $ denote the symmetric group on $n$ letters, and $ \mathbb{C}[S_n] $ its group algebra.
Let $X_n$ be the $n$-th Jucys–Murphy element $X_n = \sum_{k=1}^{n-1} (k\ n)$.
Denote by $Z_n = Z(\...
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The image of the spin group of a split real quadratic vector space under a spinor representation
Let $V$ be an even-dimensional real vector space equipped with a nondegenerate symmetric bilinear form $B$ that is split, e.g., $V = (\mathbb{R}^d)^\ast \oplus \mathbb{R}^d$ with $B((\alpha,X),(\beta,...
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