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Jun
11
comment What is known about the representation theory of the symmetric group over $\mathbb{F}_2$
There is an accepted answer already, but I should remark that over prime characteristic $p\leq n$ the group algebra is no longer semisimple, and that makes a (vague) answer to your question being "most of the Specht modules aren't even simples or semisimple, so we can't talk bout irreducible decompositions"... By the way, it is also not too difficult to determine whether a Specht is simple if you know about the block theory of symmetric group using partitions and abacus.
May
30
comment Indecomposable quiver representations
I believe someone must have some data (or a program) on this; the problem is finding the correct person to ask for it. I also think it is worth going through Assem-Simon-Skowronski book chapter VIII to see if there is a quicker way (as there are exercises in that chapter which also ask for position of $E_8$-modules with given dimension vectors). The still-developing QPA package of GAP should also be able to do a lot of computations for us, as I see that they have implemented the procedure to compute predecessor of a module in AR-quiver.
May
16
comment Left and Right minimal homomorphisms.
To partially answer your question to PavelC: No, in fact, minimality, almost splitness, these kinds of stuff are very useful for studying homological algebra, where we work with triangulated categories rather than abelian category. In fact, we usually just look at additive subcategory of some "good" category (abelian/triangulated or similar things), then tweak the original definitions.
May
16
comment Left and Right minimal homomorphisms.
For your first question, (if I am not mistaken) that was exactly Auslander was trying to play with. It's called "morphisms determined by module". I don't know much about this, but Ringel and XiaoWu Chen are looking into this very recently; and some people are generalising to other categories too. Search those keywords on arXiv/google will give you something. Sorry that I know too few to answer your question properly.
May
15
comment RHom and Koszul complexes.
How about this: let $C_\bullet := A_0\langle-i\rangle [i]$. Degree $j$ maps are built up from module hom $P_{x+j}$ to $C_x$. Becuase $hom_A(P_x,C_y)=0$ for all $y\neq i$, the degree $j$ maps from $P_\bullet$ to $C_\bullet$ are described by $hom_A(P_{j+i},C_i) = hom_A(P_{j+i},A_0\langle -i\rangle)=0$.
May
15
comment RHom and Koszul complexes.
The conclusion is for each $i\geq 0$, you have $Hom_A^\bullet(P_\bullet,A_0\langle -i\rangle[i]) = Hom_A(P_i,A_0\langle -i\rangle) = Hom_A^0(P_\bullet, A_0\langle-i\rangle[i])$. So the complex is zero in all non-zero (homological) degree.
May
14
comment RHom and Koszul complexes.
my usual notation is $Hom$ for ungraded (some people use different one), so you take morphisms in all degree, but the special nature of $A_0$ and $P_\bullet$ implies that $Hom_A(P_j,A_0\langle -i\rangle)=hom_A(P_j,A_0\langle -j\rangle)$. Now, just think about what $Hom_A^\bullet$ means, it takes morphisms of the complex $P_\bullet$ to a stalk complex (hence module) $A_0\langle-i\rangle$. So the $h$-th (homological) degree morphism is completely described by $Hom_A(P_h,A_0\langle-i\rangle[i]) = Hom_A(P_{h+i},A_0\langle -i\rangle)$.
May
14
comment RHom and Koszul complexes.
Is it clear to you that, since $P_j$ lives in (wrt Koszul grading) degree $\geq j$, we have $hom_A(P_j, A_0\langle -i\rangle)=0$ unless $i=j$?
May
13
comment What is a Complete Set of Weights of a Representation of a Lie Subalgebra?
I think a bit more context would be helpful. Can you give the full sentence/paragraph where this appears. The term string (of weights) appears in the study of crystal basis, but it may not be what your lecturer means.
May
9
comment Why are (representations of ) quivers such a big deal?
@math137, you are correct. The naive way to write down the explicity bimodule needed for the Morita equivalence would be to classify the projective indecomposable summands of the algebra up to isomorphism. This, is not an easy task in general.
May
2
comment Representation theory& module
it seems to me you are asking surjectivity of $\rho_V$. Just put $R=k$ a field, ask yourself if a map $k$-linear map $k\to End_k(V)$ with $V$ a $k$-vector space (=$k$-module) is necessarily surjective.
May
1
comment Representing natural numbers as matrices by use of $\otimes$
While I myself cannot understand why OP want to "turn natural numbers into matrices", the idea of replacing multiplication of natural numbers (or an element of an algebra) by tensor product is actually one of the key ingredient in modern representation theory, or rather categorification. In the case of of natural number, consider the category of vector spaces, then isomorphism classes of objects can be identified with (decategorify) natural numbers (being the dimension of space), while the operations: direct sum and tensor product, decategorify to addition and multiplication.
May
1
comment Representing natural numbers as matrices by use of $\otimes$
I agree people really should explain their reason(s) for downvoting on SE. I guess in this particular thread, the main reason is that both OP's question and comment are too vague on what s/he is doing, especially on the reason why you want a matrix "representation". I also double quoted representation as this is not the mathematically correct term to use from what I understood from your question, which may be another reason for the downvote. Though, this problem of misusing the term 'representation theory' on SE is not rare at all...
Apr
16
comment Schur-Weyl duality from Double Commutant Theory
A version of Schur-Weyl duality is to say $\mathbb{C}[S_n]$ and $\mathbb{C}[GL(V)]$ are each other's commutant, so that gets you (1); that also tells you your algebra is acting on the wrong side. For (2), what is your $U(V)$? Unitary group?
Apr
11
awarded  Yearling
Apr
9
comment What are the consequences of presentation of an algebra by generators and relations?
How much information the generators and relations give is really dependent on subject. For example, as Matt mentioned, if (quotiented) path algebras are essentially associative algebras with 1-dimensional simple top presented in generators (arrows of the quiver) and relations, and where one can work with their homological algebra pretty much combinatorially. On the other extreme, the Hecke algebras, universal eneveloping algebras of Lie algebras can also be defined by generators and relations, but their representations are way much harder to study compare to path algebras.
Mar
31
comment Question concerning Morita equivalence and an algebra over a field which is not algebraically closed
Have you checked the "single vertex with 2 arrows" quiver?
Mar
26
revised Finding another representation of a group represented by a set of matrices
taking out representation-theory tag
Mar
26
suggested suggested edit on Finding another representation of a group represented by a set of matrices
Mar
19
comment Generalized Schur-Weyl Duality
For $O(n)$ and $Sp(n)$, the algebra on the other side of the Schur-Weyl duality (replacing the group algebra $kS_n$) is called Brauer algebra, first studied in [H.Wenzl - On the structure of Brauer's centralizer algebras]; Goodman-Wallach's book also has an account on this subject. I don't know about general theory for reductive group though.