Fermionic Diagonal Coinvariants

Let $W$ be a complex reflection group of rank $n$ acting on its reflection representation V \cong \mb{C}^n. The doubly graded action of $W$ on the exterior algebra $\wedge (V \oplus V^*)$ induces an action on the quotient by the ideal generate by $W$-invariants with vanishing constant term \FDR_W = \wedge (V \oplus V^*) / \langle \wedge (V \oplus V^*)^W_{+} \rangle. We describe the bi-graded $W$-module structure of \FDR_W and introduce a variant of Motzkin paths that descends to the standard monomial basis of \FDR_W with respect to certain term order. The top degree of \FDR_W exhibits the Narayana refinement of Catalan numbers. When $W = S_n$, the symmetric group, \FDR_{S_n} \cong R_{n,0,2}, where $R_{n,0,2}$ is the special case of the Boson-Fermionic diagonal coinvariants with two sets of Fermionic variables. In this case, the $(i,j)$-th degree component is a difference of Kronecker product of two hook Schur functions. In addition we consider a module $M_{n,m}$ spanned by $m$-ary strings of length $n$. When $m = 2$, as a vector space, M_{n,2} \cong \mb{C}[X_n] / \langle x_1^2, \ldots, x_n^2 \rangle. The trivial component of \dr_n \otimes M_{n,2} is a weighted sum of $q,t$-Narayana numbers which is a different $q,t$-Catalan number than the alternant of \dr_n. At $t = 1$, the trivial component equals the inversion generating function for $321$-avoiding permutations

On the cohomology of irreducible holomorphically symplectic varieties

We study the cohomology of IHS varieties. After giving some results of the algebraic properties of the Fujiki relation, we proceed to a description of the integral cohomology of the generalized Kummer fourfold. We implement two models for the multiplicative structure of the cohomology ring of Hilbert schemes and give a series of computer based results in the case of Hilbert schemes of points on K3 surfaces