A Hessenberg generalization of the Garsia-Procesi basis for the cohomology ring of Springer varieties (Q612936)

Consider the full flag variety on complex \(n\)-space. Given an \(n\) by \(n\) nilpotent matrix \(X\), the Steinberg variety associated to \(X\) is the subvariety of those flags preserved by \(X\). A Hessenberg function \(h\) is a function from \(\{1, 2, \dots, n\}\) to itself such that \(h(i) \geq i\) for all \(i\) and \(h(i) \leq h(i + 1)\) for all \(i \leq n - 1\). The Hessenberg variety associated to an \(X\) and \(h\) (as above) is the subvariety of the full flag variety consisting of those flags which are preserved by \(X\) relative to \(h\). That is, for which \(X \cdot V_i \subseteq V_{h(i)}\) for all \(i\) (where \(V_1 \subseteq V_2 \subseteq \cdots \subseteq V_n = {\mathbb C}^n\) is a flag). The Springer variety corresponds to the case when the function \(h\) is the identity. Of interest here are questions related to determining the integral cohomology of such Hessenberg varieties. In the case of the Steinberg variety, the cohomology ring is known to be a quotient of a polynomial ring by the so-called Tanisaki ideal. A basis of monomials for this quotient ring was constructed by \textit{A. M. Garsia} and \textit{C. Procesi} [Adv. Math. 94, No. 1, 82--138 (1992; Zbl 0797.20012)] (which is referenced in the title of the article). In the case of more general Hessenberg varieties, \textit{J. Tymoczko} [Am. J. Math. 128, No. 6, 1587--1604 (2006; Zbl 1106.14038)] gave a combinatorial method for computing the dimensions of the cohomology groups by counting certain fillings of a Young diagram. However, beyond some special cases, there is no known presentation of the cohomology ring; something which this paper is directed towards. The author defines a map from the set of fillings (of Tymoczko) to the ring of integer polynomials in \(n\)-variables which takes a filling to a monomial. The map can be extended to a vector space map over the rationals by considering formal linear combinations of fillings and extending the polynomials to rational coefficients, and the map preserves degree. In the case of the Springer variety, the author shows that the image of the set of fillings is in fact the monomial basis of Garsia-Procesi, thus providing a direct link between the combinatoric and algebraic computations of cohomology. For a general Hessenberg function \(h\), the author restricts consideration to a regular nilpotent matrix and introduces an analogue of the Tanisaki ideal depending on \(h\). It is conjectured that the cohomology ring is isomorphic to a quotient of a polynomial ring by this ideal. It is noted that the conjecture has been confirmed for some examples with small \(n\) in the special case of Peterson varieties. For the general case of a regular nilpotent matrix and arbitrary \(h\), analogous to the Steinberg case, the author shows that the image of the fillings is precisely a monomial basis for this quotient ring, giving some further evidence for the conjecture.