Questions tagged [rt.representation-theory]
Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
7,388 questions
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Number of connected acyclic gentle tree algebras
Let $X_n$ denote the number of acyclic connected gentle tree algebras (given by quiver and admissible relations over a field) with $n$ simple modules. Those are also exactly the connected quiver ...
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symplectic resolution of K theoretic coulomb branch
Consider the type $A_n$ quiver with gauge group $G=\prod_i \mathrm{GL(V_i)}$ and representation $N=\oplus_i \mathrm{Hom(N_i, N_{i+1})}$, will the K-theoretic Coulomb branch $Spec(\mathrm{K}^{ G(\...
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Spherical representations of $(G,H)$ with $H$ not connected
Let $G$ be a compact connected semisimple Lie group.
Let $H$ be a closed subgroup of $G$ not necessarily connected.
Let us denote by $H^0$ the connected component of $H$ containing the trivial element ...
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References on descent for (triangulated) dg-categories -- For computing cones in particular
If I have a triangulated dg-category $\mathcal{C}$ (or stable $\infty$-category, if you prefer) defined over a finite field $k = \mathbb{F}_q$, I can base change that category to one defined over $\...
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Definition of a semisimple p-adic analytic group
I would like to understand what it means for a $p$-adic analytic group $G$ to be semisimple.
I am especially interested in the case when $G$ is a subgroup of $\mathrm{GL}(n,\mathbb{Q}_p)$ that is ...
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Non-vanishing of quantum cohomology
Let $(X)$ now be a smooth projective variety over $(\mathbb{C})$, and consider genus-zero Gromov–Witten invariants. Define
$[I(t,\alpha,\beta,\gamma)
\sum_{d \in H_2(X,\mathbb{Z})}
\left\langle \alpha,...
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Is there a mapping from a free group to a surface group that preserves trace equivalence?
A couple of days ago I asked this question. The answer was very helpful and it has encouraged me to rework the question.
We say two elements $u$ and $v$ in a group ($G$) are "trace equivalent&...
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Rationality of Joint Subset Coalescence Probabilities for Products of Random $n$-Cycles
Let $\mathcal{C}_n$ denote the set of all $n$-cycles in the symmetric group $\mathfrak{S}_n$, which has cardinality $(n-1)!$. Draw $\sigma$ and $\tau$ independently and uniformly from $\mathcal{C}_n$, ...
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If two elements are trace equivalent in a surface group are they trace equivalent in a free group?
We say two elements $u$ and $v$ in a group ($G$) are "trace equivalent", $u \equiv_{\operatorname{tr}} v$, if for every complex representation, $\alpha : G\rightarrow \operatorname{SL}(2,\...
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Software for calculating subrepresentations of representations of $\mathop{SL}_n(\mathbb{C})$
Fix some $n \geq 2$. Let $V \cong \mathbb{C}^n$ be the standard representation of $\mathop{SL}_n(\mathbb{C})$ and let $V^{\ast}$ be its dual. Let $W$ be a representation of $\mathop{SL}_n(\mathbb{C})...
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Simplicity of $\mathfrak{g}$-sub-modules generated by a single vector
Let $\mathfrak{g}$ be a complex semisimple Lie algebra and $V$ a finite-dimensional $\mathfrak{g}$-module. Take a highest weight element $v$ in $V$ and consider the submodule generated by $v$ that is ...
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(Homological) interpretation of the Moore penrose number
Let $A=KQ/I$ be a finite dimensional quiver algebra with connected acyclic quiver $Q$ and admissible relations $I$.
Let $W$ be the Cartan matrix of $A$ (which we can assume to be lower triangular with ...
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Fac-prime modules versus local modules over representation-finite algebras
Let $A$ be a finite-dimensional algebra over a field, and let $M$ be a finitely generated indecomposable $A$-module.
I will say that $M$ is Fac-prime if whenever
$M \in \mathrm{Fac}\{X_1,X_2\}$, then ...
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Evaluating a hook-character sum over cycle products: Pieri rule exhaustiveness and vanishing for $j > m$
I am trying to prove the following closed form. Let $C_m = (1, 2, \dots, m) \in \mathfrak{S}_{n}$ be a fixed $m$-cycle and let $\mathfrak{S}_{n-m}^{+}$ denote the symmetric group on $\{m+1, \ldots, n\}...
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How to extend a sign-reversing involution on products of $n$-cycles that exits the conjugacy class?
I am working on a combinatorial proof regarding the cycle structure of the product of two uniformly random $n$-cycles, $\sigma, \tau \in \mathcal{C}_n \subset S_n$. Specifically, I am evaluating the ...
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