# Alex Bartel

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 Name Alex Bartel Member for 2 years Seen 53 mins ago Website Location University of Warwick, UK Age 30
I am a Zeeman Lecturer at Warwick University. My research interests lie in (the intersection of) algebraic number theory and representation theory. In the former, I am interested in integral Galois module structures and in the arithmetic of elliptic curves over number fields. In the latter, I work on integral representations of finite groups, and also on the connections between the Burnside ring and the rational representation ring of a finite group.
 Apr16 awarded ● Popular Question Mar13 comment some induced characters of Lie groupsYou can compute the inner product of $\chi$ with itself, and then with $\chi'$. That will give you the result (if it is true). In order to compute these inner products, you need Frobenius reciprocity (twice), Mackey's formula, and a double coset decomposition of $P\backslash G/P$. For a concrete description of the latter, see e.g. page 7 of arxiv.org/pdf/0907.3970v1.pdf. Mar8 awarded ● Enlightened Mar8 awarded ● Nice Answer Mar6 comment Character table entries and sums of roots of unitySo here is the problem, I think: suppose that among the $m$-th roots of unity, the first two are 1, and the remaining ones are distinct primitive $m$-th roots of unity. Then the subgroup whose $S_n$-component is trivial and whose $G$-component is of the form $(x,y,1,1,\ldots, 1)\in (\mathbb{Z}/m\mathbb{Z})^n$ is the kernel of the representation you are constructing. But this subgroup is not normal. So, unless I am mistaken, what you have constructed is not a representation. Mar6 comment Character table entries and sums of roots of unityAll the irreducible representations of $S_n\ltimes G$ are obtained as follows: start with a one-dimensional representation $\chi$ of $G$, extend it trivially to $S\ltimes G$, where $S$ is the stabiliser of $\chi$ in $S_n$, take any irreducible representation $\rho$ of $S$, viewed as a representation of $S\ltimes G$. Then ${\rm Ind}^{S_n\ltimes G}\chi\otimes \rho$ is an irreducible representation, and they all arise in this way. Is the representation you have constructed somewhere among these? I don't quite see that right now. Mar6 comment Permutation character of the symmetric group on subsets of certain sizeSee also exercise 8 here: dpmms.cam.ac.uk/study/II/RepresentationTheory/…. To compute the inner product, as in the exercise, observe that under the permutation action of $S_n$ on $X_k\times X_l$, the pairs $(A,B), (C,D)\in X_k\times X_l$ lie in the same $S_n$-orbit if and only if $\#(A\cap B) = \#(C\cap D)$. Feb18 comment Subgroups of $\mathbb{Z}^n$Yemon, the question you have linked to specifically asks about not necessarily finitely generated free abelian groups. The OP does not need to read the Wikipedia proof for his question, this is much more elementary. Feb5 awarded ● Nice Question Jan17 comment sylow subgroups of GL(n,q)See in particular Derek Holt's answer to the question linked to by Jim Humphreys. Jan6 awarded ● Enlightened Jan6 awarded ● Nice Answer Jan3 revised Littlewood Richardson rule and seminormal basis of Specht modulesdeleted 16 characters in body Dec31 comment Why is the Brauer group of a local field is $\mathbb {Q/Z}$? Is it an accident?Dear Wan Lee, it is not at all clear what you are actually asking. Please try to ask a precise question that actually admits a definitive answer. See the "how to ask" link at the top, and browse around a bit to get a feel for what sorts of questions this site is intended for. Dec10 answered Does the linear representations of an finite group on an k-vector space forms a ring?