Nested Hilbert Schemes and the nested $q,t$-Catalan series
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008) (2008)
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In this paper we study the tangent spaces of the smooth nested Hilbert scheme $\mathrm{Hilb}^{n,n-1}(\mathbb{A}^2)$ of points in the plane, and give a general formula for computing the Euler characteristic of a $\mathbb{T}^2$-equivariant locally free sheaf on $\mathrm{Hilb}^{n,n-1}(\mathbb{A}^2)$. Applying our result to a particular sheaf, we conjecture that the result is a polynomial in the variables $q$ and $t$ with non-negative integer coefficients. We call this conjecturally positive polynomial as the "nested $q,t$-Catalan series,'' for it has many conjectural properties similar to that of the $q,t$-Catalan series.
@article{DMTCS_2008_special_255_a44,
author = {Can, Mahir Bilen},
title = {Nested {Hilbert} {Schemes} and the nested $q,t${-Catalan} series},
journal = {Discrete mathematics & theoretical computer science},
year = {2008},
volume = {DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008)},
doi = {10.46298/dmtcs.3636},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3636/}
}
TY - JOUR AU - Can, Mahir Bilen TI - Nested Hilbert Schemes and the nested $q,t$-Catalan series JO - Discrete mathematics & theoretical computer science PY - 2008 VL - DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008) UR - http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3636/ DO - 10.46298/dmtcs.3636 LA - en ID - DMTCS_2008_special_255_a44 ER -
%0 Journal Article %A Can, Mahir Bilen %T Nested Hilbert Schemes and the nested $q,t$-Catalan series %J Discrete mathematics & theoretical computer science %D 2008 %V DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008) %U http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3636/ %R 10.46298/dmtcs.3636 %G en %F DMTCS_2008_special_255_a44
Can, Mahir Bilen. Nested Hilbert Schemes and the nested $q,t$-Catalan series. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008) (2008). doi: 10.46298/dmtcs.3636
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