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On fusion product and N!/k-conjecture

  • Anton Khoroshkin
  • Ievgen Makedonskyi
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Abstract

We formulate a version of Schur--Weyl duality for the current Lie algebra glV[x,y]=glV⊗ℂ[x,y]. Under this duality, the Garsia--Haiman modules of the N!-conjecture become cyclic and cocyclic glV[x,y]-modules, and we describe them as iterated fusion and cofusion products of tautological glV-modules. We show that Haiman's N!-theorem for a diagram is equivalent to the coincidence of the fusion and the cofusion filtrations on the tensor product of the local Weyl modules attached to its rows, and that the fusion and cofusion products of Garsia--Haiman modules are associative. Let λ and μ be Young diagrams obtained by removing two different corners from the same diagram. We construct an iterated fusion product of S2V with local Weyl modules, which is a quotient of the common quotient of the Garsia--Haiman modules of λ and μ, fits into short exact sequences with each of them, and has Butler's intersection polynomial as its character. This gives a representation-theoretic proof of Butler's conjecture. The same construction gives lower bounds for the dimensions in the N!/k-conjecture.

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Cite

BibTeXRIS
BibTeX
@misc{khoroshkin2026fusion,
  title = {{On fusion product and N!/k-conjecture}},
  author = {Anton Khoroshkin and Ievgen Makedonskyi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.07461},
  eprint = {2610.07461},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.07461},
}
RIS
TY  - UNPB
TI  - On fusion product and N!/k-conjecture
AU  - Anton Khoroshkin
AU  - Ievgen Makedonskyi
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.07461
UR  - https://arxiv.org/abs/2610.07461
AB  - We formulate a version of Schur--Weyl duality for the current Lie algebra gl_V[x,y]=gl_V⊗ℂ[x,y]. Under this duality, the Garsia--Haiman modules of the N!-conjecture become cyclic and cocyclic gl_V[x,y]-modules, and we describe them as iterated fusion and cofusion products of tautological gl_V-modules. We show that Haiman's N!-theorem for a diagram is equivalent to the coincidence of the fusion and the cofusion filtrations on the tensor product of the local Weyl modules attached to its rows, and that the fusion and cofusion products of Garsia--Haiman modules are associative. Let λ and μ be Young diagrams obtained by removing two different corners from the same diagram. We construct an iterated fusion product of S^2V with local Weyl modules, which is a quotient of the common quotient of the Garsia--Haiman modules of λ and μ, fits into short exact sequences with each of them, and has Butler's intersection polynomial as its character. This gives a representation-theoretic proof of Butler's conjecture. The same construction gives lower bounds for the dimensions in the N!/k-conjecture.
ER  -