Abstract The superspace ring $\Omega _n$ is a rank n polynomial ring tensored with a rank n exterior algebra. Using an extension of the Vandermonde determinant to $\Omega _n$ , the authors previously defined a family of doubly graded quotients ${\mathbb {W}}_{n,k}$ of $\Omega _n$ , which carry an action of the symmetric group ${\mathfrak {S}}_n$ and satisfy a bigraded version of Poincaré Duality. In this paper, we examine the duality modules ${\mathbb {W}}_{n,k}$ in greater detail. We describe a monomial basis of ${\mathbb {W}}_{n,k}$ and give combinatorial formulas for its bigraded Hilbert and Frobenius series. These formulas involve new combinatorial objects called ordered set superpartitions . These are ordered set partitions $(B_1 \mid \cdots \mid B_k)$ of $\{1,\dots ,n\}$ in which the nonminimal elements of any block $B_i$ may be barred or unbarred.
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Lefschetz Theory for Exterior Algebras and Fermionic Diagonal Coinvariants
Abstract Let $W$ be an irreducible complex reflection group acting on its reflection representation $V$. We consider the doubly graded action of $W$ on the exterior algebra $\wedge (V \oplus V^*)$ as well as its quotient $DR_W:= \wedge (V \oplus V^*)/ \langle \wedge (V \oplus V^*)^{W}_+ \rangle $ by the ideal generated by its homogeneous $W$-invariants with vanishing constant term. We describe the bigraded isomorphism type of $DR_W$; when $W = {{\mathfrak{S}}}_n$ is the symmetric group, the answer is a difference of Kronecker products of hook-shaped ${{\mathfrak{S}}}_n$-modules. We relate the Hilbert series of $DR_W$ to the (type A) Catalan and Narayana numbers and describe a standard monomial basis of $DR_W$ using a variant of Motzkin paths. Our methods are type-uniform and involve a Lefschetz-like theory, which applies to the exterior algebra $\wedge (V \oplus V^*)$.
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- Award ID(s):
- 1953781
- NSF-PAR ID:
- 10251736
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract Let $\Theta _n = (\theta _1, \dots , \theta _n)$ and $\Xi _n = (\xi _1, \dots , \xi _n)$ be two lists of $n$ variables, and consider the diagonal action of ${{\mathfrak {S}}}_n$ on the exterior algebra $\wedge \{ \Theta _n, \Xi _n \}$ generated by these variables. Jongwon Kim and the 2nd author defined and studied the fermionic diagonal coinvariant ring$FDR_n$ obtained from $\wedge \{ \Theta _n, \Xi _n \}$ by modding out by the ideal generated by the ${{\mathfrak {S}}}_n$-invariants with vanishing constant term. On the other hand, the 2nd author described an action of ${{\mathfrak {S}}}_n$ on the vector space with basis given by noncrossing set partitions of $\{1,\dots ,n\}$ using a novel family of skein relations that resolve crossings in set partitions. We give an isomorphism between a natural Catalan-dimensional submodule of $FDR_n$ and the skein representation. To do this, we show that set partition skein relations arise naturally in the context of exterior algebras. Our approach yields an ${{\mathfrak {S}}}_n$-equivariant way to resolve crossings in set partitions. We use fermions to clarify, sharpen, and extend the theory of set partition crossing resolution.
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rnaa203
