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_Sn \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 x₁², \ldots, x_n² \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.

Author dc:creator

• Kim, Jongwon

Advisors dc:contributor.advisor

• Jim Haglund
• Julia Hartmann

Statement dc:rights

• Jongwon Kim

Language dc:language

en

Repository record dc:identifier.uri

https://repository.upenn.edu/handle/20.500.14332/32254

OAI identifier oai:identifier

oai:repository.upenn.edu:20.500.14332/32254