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

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.