On the minimal power of $q$ in a Kazhdan-Lusztig polynomial
Abstract
For $w$ in the symmetric group, we provide an exact formula for the smallest positive power $q^{h(w)}$ appearing in the Kazhdan-Lusztig polynomial $P_{e,w}(q)$. We also provide a tight upper bound on $h(w)$ in simply-laced types, resolving a conjecture of Billey-Postnikov from 2002.
- Publication:
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arXiv e-prints
- Pub Date:
- March 2023
- DOI:
- arXiv:
- arXiv:2303.13695
- Bibcode:
- 2023arXiv230313695G
- Keywords:
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- Mathematics - Combinatorics;
- Mathematics - Representation Theory
- E-Print:
- 24 pages. v2: minor corrections, to appear in Advances in Math