# Questions tagged [rt.representation-theory]

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

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### Invariants of general linear groups under torus action

Let $G=GL_n$ be the general linear group (let's say over an algebraically closed field of char $=0$). Let's denote as $T$ the torus of diagonal matrices: is there an explicit description of the ...

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### Schur's Orthogonality relations: proposition example

I'm having trouble finding an example for this proposition:
Proposition 4.2.10. Let $G$ be a finite group. Let $\varphi^{(1)}, \ldots, \varphi^{(s)}$ be a complete set of representatives of the ...

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### What is higher representation theory?

Can anyone please introduce higher representation theory?
By Yoneda embedding, we know that global dimension of finitely generated category $\bmod\Lambda$ of Artin algebra $\Lambda$ is no more than $2$...

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### What are the indecomposable modules over $\mathbb{F}_2(C_2\times C_2)$?

Let $C_2$ be the cyclic group of order $2$ and $\mathbb{F}_2$ the field with $2$ elements. Consider the group algebra $A:= \mathbb{F}_2 (C_2\times C_2)$. It is well-known that $A$ has infinite ...

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### Partial sum of Weingarten functions over symmetric group

I have a question about partial sums of Weingarten functions. The Weingarten functions are defined as
$$
E_U[U_{i_1,j_1}\dotsm U_{i_k,j_k}U^*_{i'_1,j'_1}\dotsm U^*_{i'_k,j'_k}]=\sum_{\alpha,\beta \in \...

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### On the definition of the Cherednik algebra of a variety with a finite group action

Let $X$ be a connected complex smooth affine variety, acted on by a finite group $G$. We define a reflection hypersurface $(Y,g)$ as a smooth codimension one subvariety $Y\subset X$ which is fixed by $...

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### Supercuspidal, spherical and discrete series representation

Let $G$ be an algebraic group defined over $\mathbb{Q}$ with maximal unipotent radical $N$. Let $\pi$ be an admissible representation of $G(\mathbb{Q}_p)$, we say that this representation is ...

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### Modules with special properties

Let $A$ be a finite dimensional algebra and $M$ an indecomposable (right) module with the property that every nilpotent element of $End_A(M)$ annihilates the socle $soc_A(M)$ of $M$. Note that $M$ is ...

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### Hives for other root systems?

Littlewood-Richardson coefficients $c^\lambda_{\mu\nu}$ have numerous combinatorial interpretations, including the hive model by Knutson and Tao, see here. For other root systems, there are also ...

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### Prove that $\overline{a}_{11}$ is a prime element in $R$

Consider the affine space given by four $2\times 2$ matrices, i.e., $\mathbb{A}^{16}\cong M(\mathbb{C})_{2\times 2}^4$. Now, consider the algebraic set $V$ given by the vanishing of the relation $AB-...

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### Normalizing a parameter in a regression

I am thinking about the possibility of making a parameter in my regression, let's say the $\lambda$ in a ridge regression, somehow, inside a range : $\lambda \in [0,1]$. Do you have any ideas how I ...

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### Testing whether the endomorphism ring of an $A$-module is isomorphic to $A$

Let $A=KQ/I$ be finite dimensional quiver algebra and $M$ an $A$-module.
Question: Is there a good/fast test whether we have $A \cong End_A(M)$ (or $A^{op} \cong End_A(M)$) using for example the GAP-...

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### Is there a cohomological interpretation of the bilinear form arising from Clifford theory?

For this question all groups are finite, and representations are over $\mathbb{C}$. The setup is that we have $N$ a normal subgroup of $G$, with abelian quotient $A$, and an irrep $V$ of $N$ that is ...

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### Non-rigid modules and Auslander-Reiten quiver

I have a question about proving that a module is non-rigid using Auslander-Reiten quiver. Suppose that we have some algebra $A$ and the components of its Auslander-Reiten quivers are tubes like the ...

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### Reference request on Plancherel measure for partitions whose parts differing by more than $1$

Given an (unrestricted) integer partition $\lambda$ of $n$, let $f_{\lambda}$ denote the number of standard Young tableaux (SYT) of shape the Young diagram $Y(\lambda)$ of $\lambda$. Then,
$$\sum_{\...

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### Is Theorem 4.4 in Borel[1976] also true for covering group?

Theorem 4.4 in Borel - Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup claims that the induced module $I(E)$ and coinduced module $P(E)...

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### Is this a counterexample to Reineke's conjecture on total stability conditions for Dynkin type quivers?

Let $A=KQ$ be a path algebra over a field $K$ with finite connected quiver $Q$.
A slope function $\mu$ is a function of the form $\mu=\sigma/dim$ defined on the Grothendieck group $K_0(A) \setminus 0$ ...

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### Harmonic flow on the Young lattice

Let me begin with some preliminary concepts: A positive real-valued function $\varphi: P \rightarrow \Bbb{R}_{>0}$ on a locally finite, ranked poset $(P, \trianglelefteq)$ is harmonic if
$\varphi(\...

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### How much is known about the non-degeneracy of Quiver-with-potential associated to closed punctured surfaces?

The potential of the quiver associated to surfaces is the canonical one given by Labardini-Fragoso's 2009 paper, who proved that the the QP associated to surfaces whose boundary is nonempty is rigid ...

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### Is any $n$-angulated category a $(n-2)$-cluster tilting subcategory of some triangulated category?

Geiss, Keller and Oppermann told us in "n-angulated categories" that some $(n-2)$-cluster tilting subcategory of a triangulated category is a $n$-angulated category.
$\require{wasysym}$
...

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### Invariants for the isotropy representation of a Riemannian symmetric space

Statement: Let $M = G/K$ be a Riemannian symmetric space of compact type, and $V = T_o M$ be its isotropy representation (of $K$ acting on the tangent space of $M$). Then the Hilbert–Poincaré series $...

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### Relationship between $\mathbb{S}^{\nu}V \otimes \mathbb{S}^{\lambda}(V^{*})$ and $\mathbb{S}^{\nu / \lambda}V$

For partition $\mu$ let $\mathbb{S}^{\mu}V = V^{\otimes \mu} \cdot c_{\mu}$, where $c_{\mu}$ is the Young symmetrizer. I'm trying to prove that $\mathbb{S}^{\nu / \lambda}V$ is the polynomial part of $...

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### Algorithmically handling the Spin groups in larg(ish) dimensions

Question: Is there a reasonably efficient algorithmic representation of $\mathit{Spin}_n$? By this I mean, a way to store its elements and operate on them (multiply, inverse, maybe compute ...

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### Semi simplicity over commutative algebras over non-algebraically closed fields

I have already posted this on stackexchange
I have a question:
If k is an arbitrary field then is it true that if $M$ a finite dimensional $k[x, y]$ is semisimple as a $k[x]$ module and also as a $...

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### Scalar products on symmetric functions behaving like the Macdonald scalar product

The Macdonald symmetric functions (or Macdonald polynomials)
$P_\lambda(x)$ are orthogonal with respect to the Macdonald scalar
product
$$ \langle p_\lambda,p_\mu\rangle =
\delta_{\lambda\mu}z_\...

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### Row of the character table of symmetric group with most negative entries

The row of the character table of $S_n$ corresponding to the trivial representation has all entries positive, and by orthogonality clearly it is the only one like this.
Is it true that for $n\gg 0$, ...

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### Duals of unipotent characters of classical finite groups of Lie type in terms of Lusztig's symbols

The irreducible unipotent characters of classical finite groups of Lie type have been classified by Lusztig using the combinatorical notion of "symbols", see "Irreducible ...

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### About weak integrals: Appendix of Folland's book "A course in abstract harmonic analysis"

Consider the following fragment from Folland's book "A course in abstract harmonic analysis":
All integrals are here to interpreted in the weak sense (see p285 in Folland's book). Why is ...

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### The Jacquet module of the Steinberg Representation

I have also posted this question also on Math Stack Exchange, please inform me if the level is too low for this forum.
Let $G=GL_2(F)$ where $F$ is a non-Archimedean local field of characteristic $0$, ...

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### Is a $k[G]$-module which is free on every cyclic subgroup free?

Let $G$ be a finite group and let $M$ be a representation of $G$ over a field $k$. Suppose that, for every cyclic subgroup $C$ of $G$, we have $M|_{C} \cong k[C]^{\oplus [G:C]}$. Can we conclude that $...

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### Is a subquotient of an unramified principal series unramified?

The answer to this question may be true in a much more general setting, but I ask it in the precise case I need it.
Let $G$ be the unramified unitary group $U(3)$, built out of an unramified extension ...

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### Can closure of an orbit under a reductive action contain infinitely many orbits?

I posted this on math.se a week ago, currently it has 23 views and no other feedback.
Here on MO there are several questions about orbit closures but I could not find anything about what I need.
To be ...

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### Non-isomorphic direct products of a solvable and a semisimple Lie algebra

Given a solvable Lie algebra $\frak{a}$ and a semisimple Lie algebra $\frak{g}$ we can take their semidirect product $\frak{a} \rtimes \frak{g}$, with respect to a Lie algebra map $\frak{g} \to \...

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### Counting points of parabolic Springer fibers

Let $G$ be a reductive group over an (algebraically closed ) field. To each parabolic subgroup $P \subseteq G$ and $x \in G$ we can consider two types of partial Springer fibers associated to it :
$$1)...

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### Fusion rules for the Lie algebra $\frak{so}_{2n+1}$

For the Lie algebra $\mathfrak{so}_{2n+1}$ where can I find a description of the fusion rules of it fundamental representations? In more detail: For $\pi_i$ and $\pi_j$ two fundamental weights of $\...

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### Can we use formal groups to recover Lie-theoretic representation theory in characteristic p?

In differential geometry, Lie's theorems allow us to integrate any Lie algebra representation to a Lie group representation. The algebraic version of this is more complicated (and I'm not terribly ...

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### Does the centralizer of a regular element in a semisimple Lie algebra act by polynomials?

Let $\mathfrak g$ be a semisimple Lie algebra over $\mathbb C$, $\rho : \mathfrak g \to \operatorname{End}(V)$ a finite-dimensional irreducible representation and $x \in \mathfrak g$ regular with ...

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### Characters of upper triangular matrices over finite field - reference request

Let $B_n$ be the group of upper matrices and $U_n$ the subgroup of unipotent upper triangular matrices. I would like some references which discusses complex character theory of $B_n(\mathbb{F}_q)$ for ...

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### Irreducible unitary representations of $\mathrm{SL}(n,\mathbb R)$ from those of $\mathrm{GL}(n,\mathbb R)$

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$In the case of a non-Archimedean local field $\mathbb F$, one may reduce the representation theory of $\SL(n,\mathbb F)$ to that of $\GL(n,\...

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### How to go from primitive idempotents in $\text{End}_A(M)$ to primitive idempotents in $A$?

Let $K$ be a large enough finite field, let $A$ be a finite-dimensional $K$-algebra.
Moreover, let $M$ be a finitely generated $A$-module and let $M = M_1\oplus ... \oplus M_n$ be a decomposition of $...

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### "Burnside ring" of the natural numbers and algebraic K-theory

The construction of the Burnside ring $A(G)$ of a group $G$ (usually, but not always, finite) is given by taking the Grothendieck group of the commutative semi-ring of isomorphism classes of finite $G$...

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### Projectivity of some module

Let $k$ be a algebraically closed field and suppose that $A$ and $B$ are finite dimensional $k$-algebras. If we assume that $A$ is a symmetric $k$-algebra and $A\otimes_k I$ is a projective $A\...

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### Lifting isomorphisms between linear categories

Let $C$ be a $\mathbb{Z}$-linear category, such that $C(x,y)$ is a free abelian group with finite rank, for every $x,y\in\mathrm
{Ob}(C)$. Given a commutative ring with identity $R$, let $RC$ denote ...

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### Parahoric subgroup over a local field

$\DeclareMathOperator\SL{SL}$Let $F$ be a local field and $\mathcal{O}_{F}$ its valuation ring. Let $\pi\in \mathcal{O}_{F}$ be a uniformizer and $\mathfrak{p}=\pi\mathcal{O}_{F}$. Let $G$ be a split ...

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### Are homeomorphic representations isomorphic?

Let $G$ be a finite group. Let $V_1, V_2$ be two finite-dimensional real representations. Suppose $f: V_1 \to V_2$ is a $G$-equivariant homeomorphism. Can one conclude that $V_1$ and $V_2$ are ...

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### Decomposition of tensor product of two representations of $so(10,\mathbb{C})$

There exist two 16-dimensional irreducible non-isomorphic representations of $so(10,\mathbb{C})$. Consider the tensor products of each of them with the standard (10-dimensional) representation.
What ...

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### 10-dimensional irreducible representations of $so(10,\mathbb{C})$

Are there 10-dimensional irreducible representations of the Lie algebra $so(10,\mathbb{C})$ which are not isomorphic to the standard representation?

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### What is the motivation behind symplectic/orthogonal content?

Here $\lambda'$ is the conjugate partition of $\lambda=(\lambda_1,\lambda_2,\dots)$ and cells are in the Young diagram.
The symplectic content of cell $(i,j)$ of $\lambda$ is defined by
$$c_{sp}(i,j)=\...

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### In search of a combinatorial proof on particular set of partitions

Given a partition $\lambda=(\lambda_1\geq\lambda_2\geq\dots)$, denote the conjugate partition by $\lambda'=(\lambda_1'\geq\lambda_2'\geq\dots)$. For example, if $\lambda=(4,2,2)$ then $\lambda'=(3,3,1,...

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### Reference request: ranks of quadrics containing Segre variety

Let $U_{n}$ be a vector space of dimension $n$.
From plethysm we obtain an isomorphism
$$\mathrm{Ker}(S^2(U_4\otimes U_3)^{\vee}\stackrel{p}{\rightarrow} S^2U_4^{\vee}\otimes S^2 U_3^{\vee})\simeq \...