Alex Bartel
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Registered User
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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.
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Apr 16 |
awarded | ● Popular Question |
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Mar 13 |
comment |
some induced characters of Lie groups You 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. |
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Mar 8 |
awarded | ● Enlightened |
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Mar 8 |
awarded | ● Nice Answer |
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Mar 6 |
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Character table entries and sums of roots of unity So 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. |
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Mar 6 |
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Character table entries and sums of roots of unity All 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. |
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Mar 6 |
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Permutation character of the symmetric group on subsets of certain size See 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 $\#(A\cap B) = \#(C\cap D)$. |
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Feb 18 |
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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. |
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Feb 5 |
awarded | ● Nice Question |
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Jan 17 |
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sylow subgroups of GL(n,q) See in particular Derek Holt's answer to the question linked to by Jim Humphreys. |
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Jan 6 |
awarded | ● Enlightened |
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Jan 6 |
awarded | ● Nice Answer |
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Jan 3 |
revised |
Littlewood Richardson rule and seminormal basis of Specht modules deleted 16 characters in body |
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Dec 31 |
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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. |
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Dec 10 |
answered | Does the linear representations of an finite group on an k-vector space forms a ring? |
