Questions tagged [rt.representation-theory]
Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
7,334 questions
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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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Dirichlet series that gives power of $\pi$ at positive even integer
Let $f(s)$ be a Dirichlet series with algebraic coefficients, and suppose that:
it admits a meromorphic continuation to $\mathbb{C}$;
there exists $d \in \mathbb{N}$ such that $f(2n) \in
\...
- 1,073
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Injectivity of derivations from the middle transvectant in the free Lie algebra on $\operatorname{Sym}^m$ for $\mathrm{SL}_2$
Let $G=\mathrm{SL}_2(\mathbb C)$, $V$ its standard representation, and $V_m=\operatorname{Sym}^m(V)$ with $m\equiv 2 \pmod 4$. It is classical that
$$
\Lambda^2 V_m \;\cong\; \bigoplus_{\substack{1\le ...
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How to realize this polytope as the weight polytope of a Demazure module
In $R^{2n-2}$ with basis $\Delta^+_1,\cdots,\Delta^+_{n-1},\Delta^-_1,\cdots,\Delta^-_{n-1}$, define a convex polytope with vertices $\Delta_i^++\Delta_{j}^-$ for all $0\leq i,j\leq n-1$ such that $i+...
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Identities of Bernoulli numbers from rank 2 simple Lie algebras
Denote $B_n$ as Bernoulli numbers, it is known that the following three identities hold (for $n\in \mathbb{N}$):
$$\tag{A}\label{504268_A}\frac{(2n)!}{(4n+1)!} \frac{-B_{6n+2}}{6n+2}= \sum_{k=0}^{2n} \...
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(Cohomological) Brauer group of $BG$ — reference request
Let $G$ be a smooth connected linear algebraic group over an algebraically closed field. Write $\operatorname{Br}'(BG)$ for the cohomological Brauer group of $BG$, i.e. the group of $\mathbb{G}_m$-...
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Reference request for a proof of Cayley's tree counting formula via the representation theory of the symmetric group
Inspired by a recent project Euler problem, I came up with a proof (sketched below) of Cayley's tree formula using the representation theory of $S_n$. I would like to ask for a reference in the ...
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Trivialising homogeneous vector bundles over a homogeneous space
Take the sphere $S^2$ with the standard indexing of its line bundles $E_k$, for $k$ an integer. Is it true that $E_{k} \oplus E_{-k}$ is a trivial vector bundle? If so, what is the easiest way to see ...
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Simplicity for exterior powers of $\mathfrak{g}$-modules
Let $\mathfrak{g}$ be a complex semisimple Lie algebra. Denote by $V_{\pi_m}$ the $m$-th fundamental representation of $\mathfrak{g}$. When is it true that the $k$-th exterior power of $\Lambda^k(V_{\...
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Are there infinitely many $m$ such that $S^m V$ appears in $S^{(2,1)}(S^m V)$ for $\mathrm{Sp}(V)$?
Let $V$ be the standard $2g$-dimensional representation of $\mathrm{Sp}(V)$ (with $g \ge 1$), and for each $m \ge 1$ let $S^m V$ denote the $m$-th symmetric power.
Consider the Schur functor $S^{(2,1)}...
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Does every central extension by $\mathbb{G}_m$ arise from a projective representation?
Let $G$ be a smooth connected linear algebraic group over an algebraically closed field (of characteristic zero if you wish). Let's consider (necessarily central) extensions of $G$ by the ...
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What is the significance of the height of a character?
Let $G$ be a finite group, and fix a prime $p$ which divides $|G|$.
The ordinary (complex) characters $\text{Irr}(G)$ can be partitioned into what are called $p$-blocks, and to each block $B$ can be ...
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Confusion regarding quivers with potential and cluster tilted algebras
I was looking at P. Plamondon's paper titled "Generic bases for cluster algebras from the cluster category". I'm confused about the calculation in Example 4.3. The example starts with a ...
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Rees construction for $G$-torsors
$\newcommand{\G}{\mathbb{G}}\newcommand{\A}{\mathbb{A}}\newcommand{\gr}{\operatorname{gr}}$I'm trying to understand the $\Theta$-stratification perspective on the Harder--Narasimhan stratification of $...
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Weight bases and self-dual $\mathfrak{g}$-modules
Let $\mathfrak{g}$ be a complex semisimple Lie algebra and let $V$ be a self-dual finite-dimensional representation of $\mathfrak{g}$. Assume (for simplicity) that all the weight spaces are $1$-...
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Bounding the sum of the projective dimensions of simple modules for finite dimensional algebras
Let $A=KQ/I$ be a finite dimensional connected quiver algebra with admissible ideal $I$.
Assume $A$ has finite global dimension $g$ and $n \geq 2$ simple modules. Let $t_A$ be defined as the sum of ...
- 28.2k
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Cartan matrix and graph Laplacian
The Cartan matrix for $A_n$ is almost equal (except for the diagonal entry at the endpoints) to the graph Laplacian for its Dynkin diagram. Something similar holds for the other root systems (except ...
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Rank of tensor product of irreducible representations over finite symmetric group
Let $G$ be a finite symmetric group, and let $\widehat G$ denote the set of equivalence classes of irreducible unitary representations of $G$.
For any non-trivial $\rho\in \widehat G$, we know that $\...
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Vanishing of $\mathrm H^i(X_{\mathbf P},\mathbb Z_{\ell})$ for parabolic Deligne-Lusztig varieties $X_{\mathbf P}$
Let $\mathbf G$ be a connected reductive group over $\overline{\mathbb F_p}$, and assume that $\mathbf G$ is equipped with an $\mathbb F_q$-structure induced by a Frobenius morphism $F:\mathbf G \to \...
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Why is the Huang-Lepowsky's meromorphic braided structure on $V \text{-VAMod}$ not trivial to define?
In a series of papers ``a theory of tensor products for module categories for a vertex operator algebras I-IV", Huang and Lepowsky construct [III,12.5 & IV,14.10] a meromorphic braided ...
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Coefficients of Kazhdan-Lusztig polynomials
For KL-polynomials of Schubert varieties: $$P_{\omega,y}=\sum_iq^i\dim IH^{2i}_{X_y}(\overline{X_\omega})$$
It is known that $P_{\omega,y}$ is zero unless $y\le w$ in Bruhat order. When $y=\omega$, $...
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Existence of irrational eigenvalues of a sum of representation matrices
Let $G$ be a finite non-abelian group and $H$ be a non-normal subgroup of $G$. It can be shown that there exists a non-linear irreducible unitary representation $(\rho,V)$ of $G$ such that the matrix $...
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Restriction of equivariant line bundles to a smooth component of Springer fiber being exotic sheaves
Which $G$-equivariant line bundles on $T^*(G/B)$ restricted to a given smooth component of a Springer fiber in $T^*(G/B)$ are Bezrukavnikov-Mirkovic exotic sheaves?
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The relation between Temperley-Lieb algebra and representations of $U_q \mathfrak{sl}_2$
What is a good modern reference (or maybe this is known to someone here and lacks a good reference) which explains the relation between the Temperley-Lieb algebra and representations of $U_q(\mathfrak{...
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Sum in tensor product of irreducible representations on $S_n$
Let $G$ be a finite symmetric group, and let $\widehat G$ denote the set of (equivalence classes of) irreducible unitary representations of $G$.
For any non-trivial $\rho\in \widehat G$, we know that $...
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Finding highest weight vectors in tensor products of irreducible $\mathfrak{g}$-modules
Let $\mathfrak{g}$ be a semisimple Lie algebra and let $V$ be an irreducible $N$-dimensional $\mathfrak{g}$-module. Let's denote by $e_N$ a choice of highest weight vector for $V$.
A standard problem ...
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A Coxeter group associated to finite dimensional acyclic algebras
Let $A=KQ/I$ be an acyclic quiver algebra with Cartan matrix $U$ and let $n$ be the number of vertices of $Q$.
For example when $A=KP$ is the incidence algebra of a finite poset $P$, then $U$ is just ...
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Examples of infinite length rings that admit a faithful module of finite length
What are examples of rings $R$ (associative with unit) that have the following property? There is a faithful left $R$-module $X$ of finite length and $R$, as a left $R$-module, has infinite length. ...
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Invertibility of a semi-infinite lower triangular Toeplitz matrix
Let $\left(\dots, 0, 0, a_0, a_1, a_2, \dots \right)$ be a totally positive (TP) sequence. Is its corresponding Toeplitz matrix
$$A = \begin{bmatrix}
a_0 & \cdots & \cdots & \...
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Obtaining the simple Lie algebras from finite posets
Let $P$ be a finite poset and $KP$ the $K$-vector space with formal basis the elements of $P$ where $K$ is a field. Assume here that $K$ is always of characteristic 0.
Let $U: KP \rightarrow KP$ be ...
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Characterising simple modules in the stable Auslander algebra
Let $A=KQ/I$ be a finite dimensional quiver algebra with finite connected acyclic quiver $Q$ and admissible ideal $I$. Assume furthermore that $A$ has only finitely many indecomposable modules up to ...
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References on Number Theory-oriented Representation Theory
My ongoing progress is about representation theory and number theory, to be more specific, modular representation of General linear groups over local field. My advisor ask me to submit a reading-list ...
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Irreducible representations of $S_n$
For the symmetric group $S_n$ with $n\ge 2$, there are precisely two one-dimensional irreducible representations: the trivial representation $\mathbf{1}$ and the sign representation $\text{sgn}$, ...
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Bounding the largest Fourier coefficient of $f$ minus a class function on symmetric group $S_n$
Let $G=S_n$ be a symmetric group. A class function on $G$ is any $g:G\to\mathbb C$ that is constant on conjugacy classes, i.e. $g(xy)=g(yx)$ for any $x,y\in G$. Let $\mathcal C$ denotes the set of ...
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Divided differences of TP kernels yield distinct, real-rooted polynomials
Let $\{w_j:~1\le j\le N\}$ be a set of non-zero real numbers with $\sum_{j} \frac{1}{|w_j|}<\infty$. We define a polynomial $P(\xi,z)=\sum_{k=0}^{N-1}f_s(\xi)z^{s}$, where $f_s(\xi)$ is a real ...
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Indecomposable modules over a monomial algebra
I has a question about indecomposable modules over monomial algebras.
An admissible ideal $I$ of a path algebra $kQ$ is called monomial if it is generated by some paths of length at least two. The ...
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Closeness of Schubert varieties of the affine Grassmannian for simply-connected derived group
For a connected reductive group $G$ over $\mathbb{C}$, we have the affine Grassmannian $Gr_G:=G(\mathbb{C}((t)))/G(\mathbb{C}[[t]])$ and we have the Cartan decomposition $G(\mathbb{C}((t))) = \coprod_{...
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How does $\text{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ act on $\text{Irr}(G)$?
Let $G$ be a finite group. For a field $F$ (algebraically closed of characteristic $0$), let $\text{Irr}_F(G)$ denote the irreducible characters of $G$ over $F$.
$\text{Gal}(\mathbb{C/R})$ acts on $\...
- 2,413
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Dimension of the $H$ fixed subspace of tensor product of representations
Let $G$ be a finite group and $H$ be a normal subgroup of $G$ of index $2$. Let $\operatorname{IRR}(G)$ denote the set of all inequivalent irreducible representations of $G$. For any representation $(\...
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Multiplicity of an irrep $V$ in $\Lambda^2 V$
I have a simple looking representation theory question I have been struggling with recently.
Let $V$ be a real Hilbert space and $\mathfrak g\subseteq\mathfrak so(V)$ a Lie algebra so that the action ...
- 336
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Testing whether a given quiver algebra is noetherian
Let $A=KQ/I$ with $Q$ a finite connected quiver and $I \subset J^2$ where $J$ is the ideal generated by the arrows of $Q$.
Question 1: Is there a good theory (or even a finite test) to test whether $...
- 28.2k
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Additive-equivariant vector bundles on the projective line
Let ${\mathbb G}_a = ({\mathbb C},+)$ act on ${\mathbb P}^1$ by $a \cdot [X:Y] = [X+aY:Y]$.
Question. Is the classification of ${\mathbb G}_a$-equivariant (algebraic) vector bundles on ${\mathbb P}^1$ ...
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Question on CM-finite Iwanaga-Gorenstein algebras and the (fg)-condition
Let $A$ be a finite dimensional Iwanaga-Gorenstein algebra (meaning the regular module $A$ has finite injective dimension on both sides) and assume that $A$ is CM-finite, meaning there are only ...
- 28.2k
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$\dim\left(Ra\right) = \dim\left(aR\right)$ for semisimple finite-dimensional $R$
Let $\mathbf{k}$ be a field, and let $R$ be a finite-dimensional semisimple $\mathbf{k}$-algebra. Let $a \in R$. Prove that $\dim\left(Ra\right) = \dim\left(aR\right)$, where $\dim$ denotes the ...
- 35.6k
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Conormal bundle of grassmanninan
I have a question regarding the conormal bundle of the grassmannian under the Plucker embedding
$$\mathrm{Gr}\subset \mathbb{P},$$
let me denote by $\mathcal{J}$ the ideal sheaf of the embedding.
I ...
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