# Tagged Questions

Representation theory studies (among else) representations of groups by finite matrices. The non-commutative analog of classical Fourier transforms.

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### difference between weak and strong actions on a category

If a group acts on a category (in some sense), sometimes the phrases "weak action" and "strong action" come up. I don't know what these mean though. Could someone provide an appropriate definition? (...
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### A question about unipotent characters

I have just started learning (complex) character theory of groups of Lie type and there are some misunderstandings for me in the subject. It is well-known that unipotent characters of $PGL_{n_i}(q_i)$ ...
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### Corollary of Schur's Lemma - why abelian

Corollary (of Schur's Lemma): Every irreducible complex representation of a finite abelian group G is one-dimensional. My question is now, why has the group to be abelian? As far as I know, we want ...
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### Reduced Group algebras

Take a finite group and a field of characteristic zero. The group algebra is due to Maschke's theorem semisimple so that its a finite direct sum of matrix algebras over division algebras. I like to ...
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### (Double) Coset of $GL(n, q^2)/GL(n, q)$

I am trying to understand a particular coset/double coset of the finite group $G = GL(n, q^2) = GL_n(\mathbb{F}_{q^2})$. It has a natural subgroup $H = GL(n, q)$, which can also be viewed in the ...
Denote $G = GL(2, q) = GL_2(\mathbb{F}_q)$, $B$ its Borel subgroup of upper triangular matrices, $T$ its splitting torus of diagonal matrices. The object I am interested in is $Ind_B^G\rho$, where $\... 0answers 43 views ### A kind of permutations and possible relation to cyclic groups. Any permutation that moves$n$elements in some fashion never revisiting the same until all others have been visited, in other words so that: $${\bf P}^n = {\bf I}, \text{ but no } 0<m<n \text{ ... 0answers 12 views ### Cluster algebra of finite type It is proved in the paper that a cluster algebra is of finite type if its Cartan counter part of the principal part of its seeds is a Cartan matrix of finite type. If the initial quiver of a cluster ... 1answer 392 views ### Finding a matrix representation for two Grassmann numbers. This question is more general in the sense that I want to know how one finds a particular (say matrix) representation for any object. For the case of Grassmann numbers we have from Wikipedia the ... 0answers 21 views ### What is number of irreducible characters in modular representation? In ordinary representation, I know that the number of irreducible characters is the number of conjugacy classes, what about the modular representation? Can I find a good and simple book on modular ... 1answer 29 views ### Linear representations of projective groups Does the projective linear group PSL_2(\mathbb{R}) admit faithful linear representations? In other words, does there there exist a homomorphism SL_2(\mathbb{R}) \to GL_n(\mathbb{R}), for some n, ... 1answer 18 views ### Table of e8 representations I want to understand the representation theory for the (complex-valued) e8 exceptional Lie algebra. An ideal answer to this question would contain a link to a text file (or any other format) ... 1answer 19 views ### Tensor product of representations of a Lie algebra (or Lie Superalgebra) Let V and W be finite dimensional irreducible representations of a Lie Algebra or a Lie Superalgebra. If V is one dimensional, is V\otimes W necessarily irreducible? I know this to be true ... 0answers 54 views ### Is the assignment of a root system to a semisimple Lie algebra functorial? As described here, we have a category of root systems, where a morphism from a root system \Phi in a Euclidean space E to a root system \Phi' in E' is given by a linear map f: E \to E' such ... 2answers 79 views ### Trace identities for \text{SO}(n) The Green-Schwarz mechanism in Type I string theory involves certain identities relating traces in the vector and adjoint representations of \text{SO}(n) of dimension n and n(n - 1)/2 ... 2answers 31 views ### Is the tensor product of two Yetter-Drinfeld modules a Yetter-Drinfeld module? Let U,V be two Yetter-Drinfeld modules over a bialgebra H. Is U \otimes V a Yetter-Drinfeld modules over H? Thank you very much. 1answer 29 views ### Compatibility of Yetter-Drinfeld modules. Let H be a Hopf algebra. A Yetter-Drinfeld module over H is a triple (V, \cdot, \delta), where \cdot : H \otimes V \to V , \delta : V \to H \otimes V are actions and coactions respectively, ... 1answer 111 views ### Representing natural numbers as matrices by use of \otimes What I am wanting to do is to find a unique matrix representations for Natural numbers. Say I have the number n, how can I represent this number as a matrix in which I can do matrix multiplication ... 0answers 30 views ### Commutators in the context of local Lie groups. Let G be a local Lie group in the neighbourhood V \subseteq \mathbb{C}^d with identity element denoted by e \in G. Also, let$$ t \mapsto f(t) = (f_1(t), \dots, f_d(t)) \quad \forall t \in \... 1answer 31 views ### What is the simplest example of the tame representation type? What is the simplest example of the tame representation type? I tried to find simple example could help me to understand the tame representation type. I know the definition of tame is like: A ... 0answers 45 views ### Attempt to represent gaussian integers with matrices over${\mathbb Z_+}^{4\times4}$Let us first consider the generating element for$C_2$: $$M_1 = \left[\begin{array}{cc}0&1\\1&0\end{array}\right], \text{ and } P_1 = ({M_1})^2 = I_2 = \left[\begin{array}{cc}1&0\\0&1\... 0answers 14 views ### If a\in IBr(G/N), then a\in IBr(G)? [closed] If a\in Irr(G/N), then a\in Irr(G). How about replacing Irr by IBr? 0answers 19 views ### How to compute the number of modular/Brauer characters in a p-blocks of a finite groups, for example A_5 or S_3? I do not know how to compute the modular character in a p-block of finite group.I want to know some skills for computing the number of modular characters in a p-block of a finite or some material ... 1answer 16 views ### How could I check the closedness under multiplication of the ring of symmetric functions? Let \Lambda be the ring of symmetric functions, which is defined as the subspace of the power series ring over \mathbb{C} generated by monomial symmetric functions. Now, the monomial symmetric ... 1answer 31 views ### Irreducible representations of the fundamental group of a closed surface in SU(2) For a compact Lie group G, consider the map f : G^{2n} \to G given by f(A_1, B_1, \ldots, A_n, B_n) = \displaystyle\prod_{i = 1}^{n} A_i B_i A_i^{-1} B_i^{-1} A theorem of Goldman (from the '... 1answer 59 views ### d\pi(X) is skew-symmetric. What does it mean? This is from a lemma in Lang SL_2 If \pi is a unitary representation of G, and X \in \mathfrak g, then d\pi(X) is skew symmetric on H_\pi^\infty What does skew symmetric mean here? And ... 1answer 37 views ### If H<G is abelian, and \chi(1)=[G:H] for irreducible \chi, then H contains a nontrivial normal subgroup? Suppose \chi is an irreducible character of a finite group G, and H is a nontrivial abelian subgroup such that \chi(1)=[G:H]. Why does H contain a nontrivial normal subgroup? I understand ... 0answers 32 views ### Prove V is simple \iff all non-zero vectors are cyclic I am working on some Representation Theory practice questions and I think I have given a valid proof of : Prove V \ne 0 is a simple A-Module\iff all non-zero vectors are cyclic "\leftarrow" ... 1answer 224 views ### Semisimple objects in abelian categories Let \mathcal A be any Grothendieck abelian category and 0 \neq M \in \cal A an object. It is true that M admits a simple subquotient? It is certainly true for \mathcal A=R-Mod since M ... 0answers 29 views ### Representation of A_5 Can someone give me a proper reference (a book probably)for how a 3 dimensional representation of the Alternating group A_5 is related to the reflection group H_3 or the Icosahedral group ? Thanks 0answers 64 views ### Growth of the characters of finite permutation groups in the number of symbols I have the following questions. When can the characters of the irreducible representations of the elements of a finite permutation group increase exponentially in the number of the symbols the ... 2answers 126 views ### An example of a discontinuous “\ell-adic Galois representation” Let \mathbb{F}_p be a finite filed with p elements, and G=\mathop{\mathrm{Gal}(\mathbb{F}_p^s/\mathbb{F}_p)} be its absolute Galois group. G is a pro-finite group, with the Krull topology, see ... 1answer 28 views ### Schur Multipliers in Finite Simple Groups I heard that Schur multiplier's played important role in classification of finite simple groups. By means of simple example, can one illustrate how the Schur multiplies played their role in the ... 0answers 109 views ### Reference needed for Determinant of convex combination of two matrices as a function [closed] What can one say about the function (t,A,B) \mapsto \det(tA + (1-t)B), with t \in [0,1], A, B square matrices, in my case, say, permutational matrices? Where such a function shows up? Hoping ... 2answers 324 views ### Conceptual description of the isotypical component This is probably rather simple but I have not found it in the literature. Consider the category C of representations of a finite group G, over a field k of characteristic not dividing the order ... 0answers 145 views ### Restriction of irreducible unitary representation to normal subgroup of finite index [migrated] Let G be a Lie group (or more generally a locally compact group), let N be a closed and normal subgroup of G of finite index. Let H be an infinite dimensional complex Hilbert space, and let \... 1answer 51 views ### why the algebra A = k[x]/(x^n) has finite representation type? [closed] Suppose that k is algebraically closed. Then why the algebra A = k[x]/(x^n) has finite representation type? please clarify the answer. 0answers 32 views ### does algebra over algebraically closed field has isomorphism classes of irreducible modules? Let F be an algebraically closed field and M be an F-algebra. Which are the conditions that make M has finitely many isomorphism classes of irreducible M-modules? 0answers 26 views ### Finite Dimensional Representation of Lie Algebra. Let V, W, U be finite dimensional representations of a lie algebra \mathfrak{g}. Show that \hom(V \otimes W, U) \cong \hom (V, U \otimes W^*). I think I have to use the enveloping algebra of ... 0answers 21 views ### Affine and linear reflections Let \gamma - affine reflection in complex space, which is transformation with properties: (1) \gamma is a motion (thus linear part of \gamma : \mathbf{Lin} \gamma \in U(V)), (2) \gamma ... 1answer 33 views ### Tensoring over the group ring versus tensoring over the ring in view of group representations. I was reading a chapters homology with local coefficients. Where one of the preliminary sections asks us to compute$$\mathbb{Z}_{+} \otimes_{\mathbb{Z}[\mathbb{Z}/2]}\mathbb{Z}_{-}$$Here \mathbb{... 0answers 20 views ### When H is a Yetter-Drinfeld module over itself? [closed] Let H be a bialgebra. When H is a Yetter-Drinfeld module over itself? Thank you very much. 0answers 38 views ### Singular Locus of a Schubert variety I am trying to compute the singular locus of the schubert variety X_w in G_{2,7} where w=(4,7) \in I_{2,7}. Following the notation in the book "The Grassmannian Variety: Geometric and ... 1answer 34 views ### irreducible unitary reflection group Let G be a finite irreducible unitary reflection group (i.e. without G-invariant subspaces). Given orthonomal basis, we have that g_1 \in GL(V) commutes with every element of G. It is said that ... 1answer 26 views ### About Structure of Free Algebra over K In MIT Course No. 18.712, Associative Algebra A is defined as a vector space over a field K with a bilinear associative map A \times A \to A, (a,b) \to ab. Then some examples are given, ... 2answers 503 views ### Haar Measure of a Topological Ring A topological ring is a (not necessarily unital) ring (R,+,\cdot) equipped with a topology \mathcal{T} such that, with respect to \mathcal{T}, both (R,+) is a topological group and \cdot:R\... 1answer 167 views ### Representation Theory Symmetric Group Book? I'm looking for a nice book that discusses the representation theory of the symmetric group. My background is an introductory class in representation theory. 0answers 31 views ### Name for quiver representation Let Q = (Q_0, Q_1) be a quiver, and pick some i \in Q_0. Define the quiver representation M by$$M_j = \begin{cases} k & \text{ if there is a path from$i$to$j$,} \\ 0 & \text{ ... 3answers 87 views ### Finite dimensional representations of the Weyl algebra in characteristic$p>0$I'm working through representation theory course notes of P. Etingof. In problem 1.26 it is asked to find all finite dimensional irreducible representations of the algebra$A=\frac{k[x,y]}{\left\...
On a project on how Representation theory can help improve the complexity of shape matching, I couldn't understand this result : If $V$ is the space of spherical functions, consider the ...