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arXiv · 1706.06240

Spin nilHecke algebras of classical type

Ian Johnson·Weiqiang Wang
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Abstract

We formulate and study the spin nilHecke algebras ${}^\mathfrak{b}\!{\mathrm{NH}}_n^-$ and ${}^\mathfrak{d}\!{\mathrm{NH}}_n^-$ of type B/D, which differ from the usual nilHecke algebras by some odd signs. The type B spin nilHecke algebra is a nil version of the spin type B Hecke algebra introduced earlier by the second author and Khongsap, but not for the type D one. We construct faithful polynomial representations $\mathrm{Pol}_n^-$ of the nilHecke algebras via odd Demazure operators. We formulate the spin Schubert polynomials, and use them to show that the spin nilHecke algebras are matrix algebras with entries in a subalgebra of $\mathrm{Pol}_n^-$ consisting of spin symmetric polynomials. All these results have their counterparts for the usual nilHecke algebras over the rational field. Our work is a generalization of results of Lauda and Ellis-Khovanov-Lauda in usual/spin type A.

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Ian Johnson, Weiqiang Wang. 2017-06-20. Spin nilHecke algebras of classical type. https://arxiv.org/abs/1706.06240

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