Operators on symmetric polynomials and applications in computing the cohomology of $BPU_n$

TL;DR

This paper computes the integral cohomology ring of the classifying space of the projective unitary group using symmetric polynomial operators, providing explicit structures in low dimensions and for prime-specific subgroups.

Contribution

It introduces a novel application of Young diagrams and Schur polynomials to analyze the cohomology of $BPU_n$, advancing understanding of its algebraic structure.

Findings

Determined $H^*(BPU_n; Z)$ up to dimension 11.

Identified $p$-primary subgroups of cohomology for odd primes.

Applied symmetric polynomial operators to cohomology calculations.

Abstract

This paper studies the integral cohomology ring of the classifying space $BPU_n$ of the projective unitary group $PU_n$. By calculating a Serre spectral sequence, we determine the ring stucture of $H^*(BPU_n;\mathbb{Z})$ in dimensions $\leq 11$. For any odd prime $p$, we also determine the $p$-primary subgroups of $H^i(BPU_n;\mathbb{Z})$ in the range $i\leq 2p+13$ for $i$ odd and $i\leq 4p+8$ for $i$ even. The main technique used in the calculation is applying the theory of Young diagrams and Schur polynomials to certain linear operators on symmetric polynomials.