Questions tagged [rt.representation-theory]
Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
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Finite dimensional spherical representation of $SO(n,1)(\mathbb{R})$
I'm looking for an explicit description of all the finite dimensional irreducible representation of the Lie group $SO(n,1)(\mathbb{R})$. Can you tell me, where I can find this description ? Thank you.
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some confusion about the explicit construction of irreducible representations of $S_n$
In this book chapter, the irreducible representations of the symmetric group $S_n$ is given in terms of polytabloids of a Ferrer's diagram $\lambda$, defined as
$e_t = \sum_{\pi \in C_t} \text{sgn}(\...
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Fiber functor of category of D-module on affine Grassmannian.
Geometric Satake correspondence allows us to construct Langlands dual group in a canonical way. In Mirkovic-Vilonen's paper, they prove that category of spherical perverse sheaves is an commutative ...
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Software for Planar Algebras or Group Rings
Does software exists for calculating with planar algebras or group rings? It could be part of Mathematica or be an extension of Python or Java or C. What would go into designing such a data-type ...
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Faithful characters of finite groups
Related to an answer to a previous question. The answer assume the following result:
Let $G$ be a finite group and $\rho : G \rightarrow \text{GL}(\mathbb{C}, n)$ be a faithful representation of $G$ (...
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Does the first fundamental representation of $\frak{sp}_n$ generates all the other fundamental representations
Let $\mathfrak{sp_n}$ be the symplectic Lie algebra, that is, the $C_n$ complex simple Lie algebra. Is it true that the first fundamental, which is to say the vector space, representation $V_1$ of $\...
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Enveloping algebra of affine Lie algebra is (not) noetherian
I work over an algebraically closed field of characteristic $0$. Let $\mathfrak{g}$ be a semisimple Lie algebra, $\hat{\mathfrak{g}}=\mathfrak{g}[t,t^{-1}]\oplus\mathbb{C}K\oplus\mathbb{C}D$ the ...
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Generalized Wigner 3-j symbol and Legendre functions
Let $P_{n}(x)$ the $n-th$ Legendre polynomial. It is well-knonw that $$\int_{-1}^1 P_n(x) P_m(x) P_h(x) \, dx=2\left(\begin{array}{ccc}
n & m & h\\
0 & 0 & 0
\end{array}\right)^{2}\tag{...
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Enumerating monomials in Schur polynomials
Let $s_{\lambda}(x_1,\dots,x_k)$ be the Schur polynomial associated to the partition $\lambda=(\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_k>0)$.
Among the many things involved with these ...
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Cartan matrix of the full transformation monoid ring
Let $T_n$ be the full transformation monoid of an $n$-set and $A_n=KT_n$ its monoid algebra over the complex numbers.
Question 1: Is the Cartan matrix of $A_n$ known? Im especially interested to see ...
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Finite lattices that are Koszul
Let $L$ be a finite lattice and $A=KL$ the incidence algebra of $L$.
It should be true that $L$ is modular if and only if the algebra $KL$ is quadratic (since being modular is equivalent to having no ...
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Interesting properties of "coadjoint" orbits inside $V\in \operatorname{Rep}G$
Let $G$ be a reductive group over $\mathbf{C}$. It acts on the dual of its Lie algebra $\mathfrak{g}^*$ by conjugation.
One can describe the orbits of $\mathfrak{g}^*$ explicitly (e.g. using Jordan ...
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Concept of an exact ideal of a module category
Let $R$ be a ring and $\text{Mod}\,R$ the category of (left) $R$-modules. Consider an ideal $\mathcal{I}$ of $\text{Mod}\,R$. For $R$-modules $X$ and $Y$ let $\mathcal{I}(X,Y)$ be the collection of ...
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Fixed subspaces of a family of representations $\rho_t: F_2\to GL(n,\mathbb C)$
Suppose we have a real analytic family of matrices $A_t, B_t\in GL_n(\mathbb C)$, with $t\in \mathbb R$. Suppose that for $t\in (0,1)$ there is a common non-zero eigenvector $v_t\in \mathbb C^n\...
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Frobenius algebras from symmetric polynomials
Let $K$ be a field of characteristic 0 (maybe it works for more general fields) and $K[x_1,...,x_n]$ the polynomial ring in $n$ variables. Let $e_1,e_2,...,e_n$ denote the elementary symmetric ...
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Arthur's Simple Trace Formula
In Deligne–Kazhdan–Vigneras's "Représentations des groupes réductifs sur un corps local," they use the Simple Trace Formula to prove cases of the local Jacquet–Langlands correspondence ...
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Characters on Hopf algebras
For any algebra $A$, a character for $A$ is a non-zero algebra map $c:A \to \mathbb{C}$. For $H$ be a Hopf algebra, a character is given by $\epsilon:H \to \mathbb{C}$ the counit of $H$. I am looking ...
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Construction of non-split extension of simple modules of Lie algebras using linear differential operators
Consider the natural action of $W_1=k\left\langle x,\frac{d}{dx}\right\rangle$ on $X=\mathbb C[x]$. Then $\frac{d}{dx}, x\frac{d}{dx},x^2\frac{d}{dx}$ is essentially a $\mathfrak{sl}_2$-tuple ($\left[...
- 6,158
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Obtaining quiver and relations for finite p-groups
Given a finite field $K$ with $p$ elements and a finite $p$-group $G$, is there a way to obtain the quiver and relations of $KG$ with GAP (and its package QPA)?
Since $KG$ is local, the quiver should ...
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$Hom_G(C_c^{\infty}(G),\pi)\cong Hom_{\mathbb{C}}(\pi^{\vee},\mathbb{C}) ?$
$G$ is an p-adic group, and $\pi$ is an irreducible representation of $G$, then do we naturally have
$Hom_G(C_c^{\infty}(G),\pi)\cong Hom_{\mathbb{C}}(\pi^{\vee},\mathbb{C})$? I think it is true, but ...
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Basic theorem on induction for representations of $p$-adic groups
I know a lot of places where the following is sparsely proved, but I remember there was some paper where I read it in basically the same form I write it, but unfortunately I can't remember where it ...
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Periods in the trivial extension algebra of the incidence algebra of the divisor lattice
Definition of $C_L$ for people who like number theory:
Let $m$ be a number with prime factorisation $m=p_1^{n_1} ... p_r^{n_r}$ with $n_i>0$. Define $I_m$ to be the incidence algebra of the ...
- 26.1k
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Moy-Prasad and lattice stablizers
Consider $\mathrm{SO}(5)$, or maybe $\mathrm{SO}(n)$ over your favorite locally compact non-Archimedian field of characteristic $0$. There are two interesting families of compact open subgroups. The ...
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The uniqueness of a $K$-fixed vector in a spinor representation
Consider $G=SO(2n)$ and $K=U(n)$. $(G,K)$ is a symmetric pair. I'm interested in (zonal) spherical functions on $G/K$ which are matrix elements with respect to $K$-fixed vectors in irreducible ...
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Can an amenable group have a weak mixing unitary representation without almost invariant vectors?
Does there exist a finitely generated discrete amenable group $G$ that acts on a separable Hilbert space $\mathcal{H}$ by unitary transformations, and where (1) $\mathcal{H}$ has no finite dimensional ...
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Coherent subsheaf of co-admissible modules of Schneider and Teitelbaum
Let $M$ be a co-admissible module over a Frechet Stein Algebra $A=\varprojlim A_{q_n}$ as in this paper. Let $N$ be a closed submodule of $M$. I have some difficulty in understanding lemma $3.6$ of ...
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Category of bicomodules of a cosemisimple Hopf algebra
A cosemisimple Hopf algebra $H$ is one which is equal to the direct sum of its subcoalgebras. As is well-known, this is equivalent to its category of $H$-comodules being semisimple. Is this also true ...
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Maps between symmetric powers of the natural module for $SL_2 (k)$ in prime characteristic
Let $G=SL_2(k)$ considered as a linear algebraic group over an algebraically closed field of prime characteristic. Let $E$ be the natural module for $G$ and denote by $S^r (E)$ its $r-$th symmetric ...
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Sum of skew characters over hooks and "odd" partitions
Let us call a partition odd if all its parts are odd, and let $Odd(n)$ be the set of all odd partitions of $n$, e.g. $Odd(6)=\{(5\,1),(3\, 3),(3\,1^3),(1^6)\}$.
Let $H(n)$ denote the set of all hook ...
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When is a given quiver algebra a hopf algebra?
Given a finite dimensional selfinjective quiver algebra A over a finite field (or more generally an arbitrary field). Whats the best way to check if the algebra A has a Hopf algebra structure or not? ...
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Is there a nonabelian finite simple group with Grothendieck ring of multiplicity one?
Let $G$ be a finite group. It admits finitely many irreducible complex representations $H_1, \dots, H_r$ which generate, for $\oplus$ and $\otimes$, the Grothendieck ring $\mathcal{G}(G)$ of $G$ (also ...
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How can the generators of subalgebra $\mathfrak g^{\sigma}$ of $\sigma$-stable elements be expressed through generators of Lie algebra $\mathfrak g$?
Let $\mathfrak g$ be the semisimple Lie algebra of type $D_{4}$. Let $\sigma$ be the 3-rd order automorphism of $\mathfrak g$ induced by the triality of $D_{4}$:
$$
\sigma:\alpha_{1}\mapsto\alpha_{3}\...
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Weight multiplicity formulae for $(\mathfrak g,B)$-irreps
Let $G$ be a complex reductive Lie group, $B$ a Borel subgroup, with which to define "dominant weight". Let $\lambda$ be an integral weight, not necessarily dominant, but nonetheless giving a one-...
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Special linear groups contained in symplectic groups
Let $q$ be a power of prime $p$, and $n, m, k$ positive integers such that $mk=2n$ and $2\leq m<2n$. Let $\mathrm{Sp}(2n,q)$ be the symplectic group of dimension $2n$ over $\mathrm{GF}(q)$ and $\...
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What is "special" maximal compact subgroup of algebraig group over local field?
Learning the theory of Langlands correspondence, I met the notion of "special" maximal compact subgroup of a (reductive) algebraic group over a local field.
Here, I think the word "compact" is used ...
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Jacquet module for Lie algebras?
Let $G$ be a reductive group and let $P$ be a parabolic subgroup with Levi $L$ and unipotent radical $N$. Let $F$ be a finite or local non-Archimedean field. The Jacquet Functor is a functor from $G(F)...
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Categorified versions of Mackey's functor
I would like to ask for possible references for the following very general situation, a categorified version of Mackey functors.
The question is if there are other known constructions to associate to ...
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Representations of reductive groups over local fields through parahoric induction
Let me take $G$ to be a simple (connected) split reductive group over a local field $K$. One way I might go about constructing a (smooth, admissible) complex representation $\sigma$ of $G$ is as ...
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Kazhdan-Lusztig C-basis and categorification
Does anyone know if there exists some natural way to interpret the Kazhdan-Lusztig C-basis in a categorification of the Hecke algebra ? The category of Soergel bimodules categorifies the C'-basis but ...
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reference for list of left-regular representations of real associative algebras
Suppose $\mathcal{A}$ is a unital associative algebra over $\mathbb{R}$. If we identify $\mathcal{A} = \mathbb{R}^n$ then the $\mathcal{A}$ multiplication corresponds to particular linear maps on $\...
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Rational automorphisms of semisimple algebraic groups
Suppose $G$ is a semisimple algebraic group defined over a field $k$. Let $\mathrm{Aut}(G)$ and $\mathrm{Inn}(G)$ denote the groups of automorphisms and inner automorphisms (respectively) of $G$. ...
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Other than SU(3), SO(4), SU(2)xU(1), are there compact semisimple Lie groups which exactly two 3-dimensional representations that are dual to each other?
In my original question, I asked which compact Lie groups $G$ have a certain property. Jim and Dan showed that this property is equivalent to $G$ having exactly two irreducible 3-dimensional ...
user2529
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Finite dimensional homogeneous spaces of $Diff(S^1)$
This question is a refined version of Representations of infinite dimensional Lie algebras as vector fields on manifolds
I'm interested in the finite dimensional homogeneous spaces of $Diff(S^1)$. ...
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Restriction of representation from GL(n) to O(n)
So my question is somewhat similar to Restriction from $\mathfrak{gl}_{2n}$ to $\mathfrak{sp}_{2n}$; but I was having difficulty understanding the formula given in reference (Harris & Fulton) ...
- 2,186
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Self-injective basic algebras
Do you know of any self-injective basic algebra $A$ over a field $k$ such that its enveloping algebra $A^{\mathrm{op}}\otimes_k A$ is not self-injective?
The algebra $A$ cannot be finite-dimensional, ...
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Exceptional collections with many Exts
Background definitions:
Let $D$ be a triangulated category arising in nature (for instance as the cohomology category of a dg category). An object $E$ in $D$ is called exceptional if $RHom(E,E)$ is ...
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Symmetric matrices as a module over the skewsymmetric ones
I'm trying to understand the Cartan decomposition of a semisimple Lie algebra, $\mathfrak g=\mathfrak k \oplus \mathfrak p$, where $[\mathfrak k,\mathfrak p] \subseteq \mathfrak p$, cf. the wikipedia ...
- 4,210
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What do representations of infinite-dimensional Heisenberg groups look like?
I'm interested in an infinite dim'l Heisenberg group associated to the vector space $V = L\mathbb{C}/\mathbb{C}$ = {$f \colon S^1 \to \mathbb{C}$|$f$ smooth}/(const. maps). The group is $\mathbb{C}^\...
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When does an irreducible representation remain irreducible after restriction to a semi-simple subgroup?
I suppose this question is probably elementary for experts, but I'd like to present my arguments, about which I have some doubts, and see if they are correct, or if corrections and improvements are ...
- 1,305
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Representations of products of groups (and monoids)
I have very little knowledge of representation theory, but the following has come up in my summer undergrad research project (relates to conformal field theory and geometric function theory).
Suppose ...
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