Dimensions and maximal-dimensional components of higher commuting varieties of symmetric pairs
Abstract
Let p be the (−1)-eigenspace of a complex semisimple symmetric pair. For every r≥2, we prove dim Cr(𝒩(p))≤ ⌊ rdimp/2⌋ and show that every irreducible component is isotropic for a natural family of alternating forms. For a nonzero Hermitian pair with g simple and p=p-⊕p+, put d=dimp+ and ℓ=rk(g,k). Then dim Cr(𝒩(p))=rd, dim Cr(p)=max{rd, 2d+(r−1)ℓ}. In this case we also determine the maximal-dimensional components of Cr(p); for r≥3, the only maximal-dimensional nilpotent components are (p+)r and (p-)r. For arbitrary symmetric pairs, dim Cr(𝒩(p))=ra+m for all large r, where a is the largest dimension of an abelian subspace of p consisting of nilpotent elements and m is the dimension of the variety of such subspaces. Applications give dimension and component formulas for equivariant punctual Quot schemes under explicit framing conditions, and bounds for punctual Gorenstein loci refined by socle parity.
Related
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BibTeX
@misc{pattanayak2026dimensions,
title = {{Dimensions and maximal-dimensional components of higher commuting varieties of symmetric pairs}},
author = {Santosha Pattanayak},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17737},
eprint = {2610.17737},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17737},
note = {CC0},
}
RIS
TY - UNPB
TI - Dimensions and maximal-dimensional components of higher commuting varieties of symmetric pairs
AU - Santosha Pattanayak
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17737
UR - https://arxiv.org/abs/2610.17737
AB - Let p be the (−1)-eigenspace of a complex semisimple symmetric pair. For every r≥2, we prove dim C_r(𝒩(p))≤ ⌊ rdimp/2⌋ and show that every irreducible component is isotropic for a natural family of alternating forms. For a nonzero Hermitian pair with g simple and p=p^-⊕p^+, put d=dimp^+ and ℓ=rk(g,k). Then dim C_r(𝒩(p))=rd, dim C_r(p)=max{rd, 2d+(r−1)ℓ}. In this case we also determine the maximal-dimensional components of C_r(p); for r≥3, the only maximal-dimensional nilpotent components are (p^+)^r and (p^-)^r. For arbitrary symmetric pairs, dim C_r(𝒩(p))=ra+m for all large r, where a is the largest dimension of an abelian subspace of p consisting of nilpotent elements and m is the dimension of the variety of such subspaces. Applications give dimension and component formulas for equivariant punctual Quot schemes under explicit framing conditions, and bounds for punctual Gorenstein loci refined by socle parity.
ER -