A Hessenberg Generalization of the Garsia-Procesi Basis for the Cohomology Ring of Springer Varieties

Mbirika, Aba

doi:10.37236/425

Cited by 21 publications

(28 citation statements)

References 15 publications

(43 reference statements)

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“…For each 1 < q ≤ n let q−1 be the number of dimension pairs (p, q) of T as in Definition 3.2. Mbirika showed the map T → n i=2 x i−1 i from row-strict tableaux to monomials is an injection that surjects onto a set of monomials defined by Garsia and Procesi [M,Theorem 2.2.9]. Each Schubert point w T is uniquely determined by the numbers q−1 for 2 ≤ q ≤ n so the claim follows.…”

Section: Schubert Points and Combinatorial Results About Springer Permentioning

confidence: 92%

“…is an element in their monomial basis for the cohomology of B X where X is a nilpotent matrix with Jordan blocks of size λ [GP,Proposition 4.2]. Let T denote the row-strict tableau of shape λ associated to this monomial by Mbirika [M,Theorem 2.2.9]. Lemma 3.4 thus gives w T = w .…”

Section: Schubert Points and Combinatorial Results About Springer Permentioning

confidence: 99%

“…Throughout this section we consider each row of λ[i] to be labeled with a simple reflection in decreasing order. Our labeling of the rows of λ[i] is inspired and informed by Mbirika's arguments in [M,Section 2.2]. We label the second row of λ[i] by s i−1 , the third row by s i−2 , the fourth row by s i−3 , and so on, with the top row labeled e. The string i is associated to a subset of boxes, namely those at the end of the rows in λ[i] corresponding to the simple reflections in i .…”

Section: The Two Column Casementioning

confidence: 99%

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Springer fibers and Schubert points

Precup

Tymoczko

2019

European Journal of Combinatorics

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Springer fibers are subvarieties of the flag variety parametrized by partitions; they are central objects of study in geometric representation theory. Schubert varieties are subvarieties of the flag variety that induce a well-known basis for the cohomology of the flag variety. This paper relates these two varieties combinatorially. We prove that the Betti numbers of the Springer fiber associated to a partition with at most three rows or two columns are equal to the Betti numbers of a specific union of Schubert varieties.

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Section: Schubert Points and Combinatorial Results About Springer Permentioning

confidence: 92%

Section: Schubert Points and Combinatorial Results About Springer Permentioning

confidence: 99%

Section: The Two Column Casementioning

confidence: 99%

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Springer fibers and Schubert points

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Tymoczko

2019

European Journal of Combinatorics

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“…With these preliminaries in place, we begin our computation of the Hilbert series F (H * S (Hess(N, h)), s) of the equivariant cohomology ring H * S (Hess(N, h)). Our first step towards this goal is to compute the Hilbert series of the ordinary cohomology ring H * (Hess(N, h)) using results of Mbirika [33] (we also took inspiration from related work of Peterson and Brion-Carrell as in [7]). (Hess(N, h)) (equipped with the usual grading) is F (H * (Hess(N, h)…”

Section: Hilbert Seriesmentioning

confidence: 99%

“…, x k ) the degree-β complete symmetric polynomial in the listed variables. Following [33] we define J h to be the ideal in Q[x 1 , . .…”

Section: Hilbert Seriesmentioning

confidence: 99%

The Cohomology Rings of Regular Nilpotent Hessenberg Varieties in Lie Type A

Abe

Harada

Horiguchi

et al. 2017

International Mathematics Research Notices

137

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Abstract. Let n be a fixed positive integer and h : {1, 2, . . . , n} → {1, 2, . . . , n} a Hessenberg function. The main results of this paper are twofold. First, we give a systematic method, depending in a simple manner on the Hessenberg function h, for producing an explicit presentation by generators and relations of the cohomology ring H * (Hess(N, h)) with Q coefficients of the corresponding regular nilpotent Hessenberg variety Hess(N, h). Our result generalizes known results in special cases such as the Peterson variety and also allows us to answer a question posed by Mbirika and Tymoczko. Moreover, our list of generators in fact forms a regular sequence, allowing us to use techniques from commutative algebra in our arguments. Our second main result gives an isomorphism between the cohomology ring H * (Hess(N, h)) of the regular nilpotent Hessenberg variety and the Sn-invariant subring H * (Hess(S, h)) Sn of the cohomology ring of the regular semisimple Hessenberg variety (with respect to the Sn-action on H * (Hess(S, h)) defined by Tymoczko). Our second main result implies that dim Q H k (Hess(N, h)) = dim Q H k (Hess(S, h)) Sn for all k and hence partially proves the Shareshian-Wachs conjecture in combinatorics, which is in turn related to the well-known Stanley-Stembridge conjecture. A proof of the full Shareshian-Wachs conjecture was recently given by Brosnan and Chow, but in our special case, our methods yield a stronger result (i.e. an isomorphism of rings) by more elementary considerations. This paper provides detailed proofs of results we recorded previously in a research announcement.

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Geometry of Hessenberg varieties with applications to Newton–Okounkov bodies

Abe

DeDieu

Galetto

et al. 2018

Sel. Math. New Ser.

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In this paper, we study the geometry of various Hessenberg varieties in type A, as well as families thereof. Our main results are as follows. We find explicit and computationally convenient generators for the local defining ideals of indecomposable regular nilpotent Hessenberg varieties, allowing us to conclude that all regular nilpotent Hessenberg varieties are local complete intersections. We also show that certain flat families of Hessenberg varieties, whose generic fibers are regular semisimple Hessenberg varieties and whose special fiber is a regular nilpotent Hessenberg variety, have reduced fibres. In the second half of the paper we present several applications of these results. First, we construct certain flags of subvarieties of a regular nilpotent Hessenberg variety, obtained by intersecting with Schubert varieties, with well-behaved geometric properties. Second, we give a computationally effective formula for the degree of a regular nilpotent Hessenberg variety with respect to a Plücker embedding. Third, we explicitly compute some Newton-Okounkov bodies of the two-dimensional Peterson variety.

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A Hessenberg Generalization of the Garsia-Procesi Basis for the Cohomology Ring of Springer Varieties

Cited by 21 publications

References 15 publications

Springer fibers and Schubert points

Springer fibers and Schubert points

The Cohomology Rings of Regular Nilpotent Hessenberg Varieties in Lie Type A

Geometry of Hessenberg varieties with applications to Newton–Okounkov bodies

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