Spin nilHecke algebras of classical type

We formulate and study the spin nilHecke algebras ${}^{\mathfrak{b}}\text{\hspace{0.17em}⁣}{\mathrm{N}\mathrm{H}}_n^{-}$ and ${}^{\mathfrak{d}}\text{\hspace{0.17em}⁣}{\mathrm{N}\mathrm{H}}_n^{-}$ 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 ${\mathrm{P}\mathrm{o}\mathrm{l}}_n^{-}$ 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 ${\mathrm{P}\mathrm{o}\mathrm{l}}_n^{-}$ 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.