The center conjecture for cyclotomic Hecke algebras and cohomology of quiver varieties
Abstract
For a quiver without edge loops, we construct an isomorphism from the cohomology of a Nakajima quiver variety with coefficients in an arbitrary field to the Jucys--Murphy center of the corresponding cyclotomic KLR algebra over that field. In affine type A(1)e−1, combining this with a dimension count shows that this Jucys--Murphy center coincides with the full center in characteristic $0$, when the characteristic does not divide e, and when the characteristic equals e. This follows from a geometric theorem showing that the sum of these cohomology rings is cocyclic as a module over an integral form of a loop algebra. As a consequence, we prove that the center of a cyclotomic Hecke algebra of type G(r,1,n) with q≠ 1 over any field is generated by symmetric polynomials in its Jucys--Murphy elements, and the same holds over any commutative domain in which q−1 is invertible. We also prove a generalization to the integral case of the conjecture of Krylov, Paegelow and Shlykov identifying the equivariant K-theory of the core of a Gieseker variety with the cocenter of the generic cyclotomic Hecke algebra, including the natural extension of this conjecture to positive characteristic.
Related
- Du Bois cohomology of noncommutative spaces and the Hodge numbers invariance conjecture for singular varieties
- Igusa stacks and the cohomology of Shimura varieties
- String functions and cyclotomic Hecke algebras
- Completed cohomology of Hilbert modular varieties below middle degree
- Cohomology of integral structure sheaves for open varieties
BibTeX
@misc{hu2026center,
title = {{The center conjecture for cyclotomic Hecke algebras and cohomology of quiver varieties}},
author = {Jun Hu and Andrew Mathas and Lei Shi and Ben Webster},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25101},
eprint = {2610.25101},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25101},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The center conjecture for cyclotomic Hecke algebras and cohomology of quiver varieties
AU - Jun Hu
AU - Andrew Mathas
AU - Lei Shi
AU - Ben Webster
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25101
UR - https://arxiv.org/abs/2610.25101
AB - For a quiver without edge loops, we construct an isomorphism from the cohomology of a Nakajima quiver variety with coefficients in an arbitrary field to the Jucys--Murphy center of the corresponding cyclotomic KLR algebra over that field. In affine type A^(1)_e−1, combining this with a dimension count shows that this Jucys--Murphy center coincides with the full center in characteristic $0$, when the characteristic does not divide e, and when the characteristic equals e. This follows from a geometric theorem showing that the sum of these cohomology rings is cocyclic as a module over an integral form of a loop algebra. As a consequence, we prove that the center of a cyclotomic Hecke algebra of type G(r,1,n) with q≠ 1 over any field is generated by symmetric polynomials in its Jucys--Murphy elements, and the same holds over any commutative domain in which q−1 is invertible. We also prove a generalization to the integral case of the conjecture of Krylov, Paegelow and Shlykov identifying the equivariant K-theory of the core of a Gieseker variety with the cocenter of the generic cyclotomic Hecke algebra, including the natural extension of this conjecture to positive characteristic.
ER -