1,032 reputation
38
bio website
location
age
visits member for 2 years, 5 months
seen 15 hours ago

Sep
16
comment Computing quotient representations and Hom set fort wo representations
Thanks! Now should be OK.
Sep
16
revised Computing quotient representations and Hom set fort wo representations
corrected edit
Sep
15
comment Computing quotient representations and Hom set fort wo representations
@Julian, thanks! This is embarrassing...Anyway, I have corrected the answer.
Sep
15
revised Computing quotient representations and Hom set fort wo representations
corrected calculation for char K not equals to 2
Sep
14
answered Computing quotient representations and Hom set fort wo representations
Sep
12
comment Direct sum decomposition of weight spaces and relation to Tensor products.
A small comment about answer to (Q3) as both answers didn't mention it. The (forumla for) decomposition of $V\otimes W$ for $V,W$ irreducible is usually called Clebsch-Gordan decomposition.
Jul
28
revised Can any $\theta \in \text{Hom}(S^\lambda,M^\mu)$ be written as $\theta = \kappa_t$?
corrected typeset
Jul
28
suggested suggested edit on Can any $\theta \in \text{Hom}(S^\lambda,M^\mu)$ be written as $\theta = \kappa_t$?
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.