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Asymptotic q,t-Fuss--Catalan numbers for type B

  • Alexei Oblomkov
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Abstract

Let W=W(Bn) act diagonally on h⊕h*, let S=ℂ[h⊕h*], let J⊂ S be the ideal generated by the W-alternating polynomials and mS is the maximal ideal of the origin. For sufficiently large m we compute q,t-Fuss-Catalan polynomial Cat(m)(Bn;q,t):=Hilb(Jm/(mS Jm))det−part and imply Cat(m)(Bn;1,1)=binomn(m+1)n. For proofs, we work with the Γ-equivariant Hilbert scheme Yn=nΓ-Hilb(ℂ2), Γ=μ2 and Haiman-type Koszul complex that defines the punctual locus of Yn. Our formula for Cat(m)(Bn;q,t) is derived from a localization computaion for the Haiman-type Koszul complex.

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Cite

BibTeXRIS
BibTeX
@misc{oblomkov2026asymptotic,
  title = {{Asymptotic $q,t$-Fuss--Catalan numbers for type $B$}},
  author = {Alexei Oblomkov},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.11691},
  eprint = {2609.11691},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11691},
}
RIS
TY  - UNPB
TI  - Asymptotic q,t-Fuss--Catalan numbers for type B
AU  - Alexei Oblomkov
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.11691
UR  - https://arxiv.org/abs/2609.11691
AB  - Let W=W(B_n) act diagonally on h⊕h^*, let S=ℂ[h⊕h^*], let J⊂ S be the ideal generated by the W-alternating polynomials and m_S is the maximal ideal of the origin. For sufficiently large m we compute q,t-Fuss-Catalan polynomial Cat^(m)(B_n;q,t):=Hilb(J^m/(m_S J^m))_det−part and imply Cat^(m)(B_n;1,1)=binomn(m+1)n. For proofs, we work with the Γ-equivariant Hilbert scheme Y_n=nΓ-Hilb(ℂ^2), Γ=μ_2 and Haiman-type Koszul complex that defines the punctual locus of Y_n. Our formula for Cat^(m)(B_n;q,t) is derived from a localization computaion for the Haiman-type Koszul complex.
ER  -