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bio website iaz.uni-stuttgart.de/…
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Post-Doc working in representation theory


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2d
awarded  Revival
Oct
12
comment Suppose $A$ is an invertible matrix. Prove that $det(A^{-2}) = 1/(det(A))^2$
Yes, if you already know that $\operatorname{det}(AB) =\operatorname{det}(A)\operatorname{det}(B)$.
Oct
10
answered what is dimension of orthogonal complement of a subspace of a vector space.
Oct
5
comment Let $V=V_{1}⊕ ⋯ ⊕ V_{n}$ be semisimple. $U$ irreducible. Show that $\dim_{k} (Hom_A(U,V)) $ is equal to the number of $V_i$ equivalent to $U$.
possible duplicate of Remark 3.1.3 from Introduction to Representation Theory from Pavel Etingof
Oct
3
comment Let $A=k[x]$ and let $V=k[x]/\big((x-λ )^{n} \big)$ for some $λ \in k$ and $n\in \Bbb{N}$. Then $V$ is indecomposable.
If you don't feel comfortable with the internal direct sum, just replace it with the ordinary sum: $U+U':= \{u+u': u\in U, u'\in U'\}$. The argument woks just as fine.
Oct
1
comment Let $A=k[x]$ and let $V=k[x]/\big((x-λ )^{n} \big)$ for some $λ \in k$ and $n\in \Bbb{N}$. Then $V$ is indecomposable.
$I\oplus I$ is not an internal direct sum, it has a non-trivial intersection.
Oct
1
answered Let $A=k[x]$ and $V=k[x]/((x-λ)^n)$. Find a filtration $V=V_0 ⊃ V_1 ⊃ \dots⊃ V_n=0$ such that the subsequent quotients $V_{i-1}/V_i$ are irreducible.
Oct
1
answered Let $A=k[x]$ and let $V=k[x]/\big((x-λ )^{n} \big)$ for some $λ \in k$ and $n\in \Bbb{N}$. Then $V$ is indecomposable.
Sep
30
awarded  Explainer
Sep
22
answered Show that if $V$ is isomorphic to $A/I$ for some left ideal $I$, then $V$ is a cyclic representation of $A$ over $k$
Sep
22
comment Show that if $V$ is isomorphic to $A/I$ for some left ideal $I$, then $V$ is a cyclic representation of $A$ over $k$
Related to: math.stackexchange.com/q/280685/15416
Sep
21
comment Cyclic representation
@Badshah It is a convention. Compare to if you write $K^n$ for $K$ a field. You also will take the obvious vector space structure and not some exotic which you could maybe also define.
Sep
21
comment Cyclic representation
@Badshah Of course, there could be other module structures on the vector space $A/I$. But if one writes $A/I$ for a left ideal $I$ it is implicitly assumed that the action is given by left multiplication. The right multiplication does not give you a left module in general.
Sep
21
comment Cyclic representation
@Badshah I edited to include another hint.
Sep
21
revised Cyclic representation
Added a hint for the converse
Sep
18
comment The indecomposable projective A-modules
@Dan I have expanded my answer.
Sep
15
comment Computing quotient representations and Hom set fort wo representations
@Aaron Sorry to correct you again, but $M/M_2\subset M/M_1$ is incorrect. What is true (and I guess this is what you mean) is $M/M_1\twoheadrightarrow M/M_2$. Also in the proof that $M_1$ and $M_2$ are the only proper subrepresentations, one should have an argument that $\langle \lambda (v_1-v_2)+\mu (v_1+v_2)\rangle$ cannot be a proper invariant subspace.
Sep
15
reviewed Approve suggested edit on general solution of $xy''+(1-2x)y'+(x-1)y=e^x$
Sep
15
reviewed Approve suggested edit on Homogeneous or linear?
Sep
15
comment Computing quotient representations and Hom set fort wo representations
@Dan $Hom(M,M)$ is a direct sum of a matrix ring of size $1$ (corresponding to the map $Me_1\to Me_1$) and a matrix ring of size $2$ (corresponding to the map $Me_2\to Me_2$),