#### Room 326

#### Counting anti-invariant subspaces in terms of invariant ones

#### Amritanshu Prasad

##### IMSc

*Let T be a linear operator on a finite dimensional vector space V. A subspace W of V is said to be T-invariant (resp. T-anti-invariant) if TW is contained in W (resp. dim(W+TW)=2dim(W)).*

Invariant subspaces for any operator T are easy to characterize, and in the case where V is finite, they are easy to enumerate. On the other hand, it is hard to understand anti-invariant subspaces.

In this talk, I will present a formula that enumerates T-anti-invariant subspaces of any given dimension in terms of the number of invariant ones.

A byproduct of our results will be a finite-field interpretation of the entries of the Catalan triangle of q-Hermite orthogonal polynomials, and a new proof of Touchard's formula for these entries.

Based on joint work with Samrith Ram available at arxiv.org/abs/2304.13947.

Done