The mod 2 homology of the simplex and representations of symmetric groups

Add 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:

• Algebra seminar
• School of Mathematics events
• Watson Building, Lecture Room A

Note that ex-directory lists are not shown.