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Sign components of diagonal superspace coinvariants
Abstract
We prove the sign-isotypic components of the coinvariant rings Rn(2,1) and Rn(2,0) ⊗ Rn(0,1) are isomorphic and show that the triply-graded multiplicity of this sign character is the Schröder polynomial Sn(q,t,a), divided by 1+a. This settles the sign-character component of a conjecture of Zabrocki (2019) on a module for the Delta theorem and proves a conjecture of F. Bergeron (2020) on the multiplicity of the sign character of Rn(2,1). Finally, using a result of Hogancamp (2017), we enhance a recent result of Gorsky--Mellit (2026) which relates the Khovanov--Rozansky homology of the (n,n+1)-torus knot to Rn(2,0) ⊗ Rn(0,1), by showing that the associated Poincaré series for this knot can be computed from the sign component of Rn(2,1).
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BibTeX
@misc{gonzalez2026sign,
title = {{Sign components of diagonal superspace coinvariants}},
author = {Nicolle González and John Lentfer and Hanna Mularczyk},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23297},
eprint = {2609.23297},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23297},
}
RIS
TY - UNPB
TI - Sign components of diagonal superspace coinvariants
AU - Nicolle González
AU - John Lentfer
AU - Hanna Mularczyk
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23297
UR - https://arxiv.org/abs/2609.23297
AB - We prove the sign-isotypic components of the coinvariant rings R_n^(2,1) and R_n^(2,0) ⊗ R_n^(0,1) are isomorphic and show that the triply-graded multiplicity of this sign character is the Schröder polynomial S_n(q,t,a), divided by 1+a. This settles the sign-character component of a conjecture of Zabrocki (2019) on a module for the Delta theorem and proves a conjecture of F. Bergeron (2020) on the multiplicity of the sign character of R_n^(2,1). Finally, using a result of Hogancamp (2017), we enhance a recent result of Gorsky--Mellit (2026) which relates the Khovanov--Rozansky homology of the (n,n+1)-torus knot to R_n^(2,0) ⊗ R_n^(0,1), by showing that the associated Poincaré series for this knot can be computed from the sign component of R_n^(2,1).
ER -