University of Birmingham > Talks@bham > Algebra seminar > The mod 2 homology of the simplex and representations of symmetric groups

## The mod 2 homology of the simplex and representations of symmetric groupsAdd to your list(s) Download to your calendar using vCal - Mark Wildon
- Thursday 17 May 2018, 16:00-17:00
- Watson Building, Lecture Room A.
If you have a question about this talk, please contact Chris Parker. Note Thursday Abstract: The k-faces of a (n-1)-dimensional simplex correspond to k-subsets of {1,...,n}. These subsets are permuted transitively by the symmetric group S_n. The boundary maps from simplicial homology, defined with mod 2 coefficients, give homomorphisms between the corresponding permutation modules. In recent work I consider the generalized boundary maps, defined by jumping down by two or more dimensions at once. These give ‘higher’ homology groups, affording a family of intriguing representations of S_n. In my talk, I will characterize when the homology is zero. The special case of two-step boundary maps gives a new construction of the basic spin representations of the symmetric groups. We will see that the corresponding chain complex categorifies the binomial coefficient identity $\binom{4m}{0} – \binom{4m}{2} \binom{4m}{4} – \cdots \binom{4m}{4m} = (-2)^m$. I will end with some much deeper identities that, conjecturally, are categorified by an extension of these results to odd characteristic. This talk is part of the Algebra seminar series. ## This talk is included in these lists:Note that ex-directory lists are not shown. |
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