nilHecke Algebra Research Articles

Graded dimensions and monomial bases for the cyclotomic quiver Hecke superalgebras

In this paper we derive a closed formula for the (Z×Z2)-graded dimension of the cyclotomic quiver Hecke superalgebra RΛ(β) associated to an arbitrary Cartan superdatum (A,P,Π,Π∨), polynomials (Qi,j(x1,x2))i,j∈I, β∈Qn+ and Λ∈P+. As applications, we obtain a necessary and sufficient condition for which e(ν)≠0 in RΛ(β). We construct an explicit monomial basis for the bi-weight space e(ν˜)RΛ(β)e(ν˜), where ν˜ is a certain specific n-tuple defined in (1.4). In particular, this gives rise to a monomial basis for the cyclotomic odd nilHecke algebra. Finally, we consider the case when β=α1+α2+⋯+αn with α1,⋯,αn distinct. We construct an explicit monomial basis of RΛ(β) and show that it is indecomposable in this case.

Clifford-symmetric polynomials

Based on the NilHecke algebra , the odd NilHecke algebra developed by Ellis, Khovanov and Lauda, and on Kang, Kashiwara and Tsuchioka’s quiver Hecke superalgebra, we develop the Clifford Hecke superalgebra as another super-algebraic analogue of . We show that there is a notion of symmetric polynomials fitting in this picture, and we prove that these are generated by an appropriate analogue of elementary symmetric polynomials, whose properties we shall discuss in this text.

Affine nil-Hecke algebras and quantum cohomology

Let G be a compact, connected Lie group and T⊂G a maximal torus. Let (M,ω) be a monotone closed symplectic manifold equipped with a Hamiltonian action of G. We construct a module action of the affine nil-Hecke algebra Hˆ⁎S1×T(LG/T) on the S1×T-equivariant quantum cohomology of M, QHS1×T⁎(M). Our construction generalizes the theory of shift operators for Hamiltonian torus actions [46,40]. We show that, as in the abelian case, this action behaves well with respect to the quantum connection. As an application of our construction, we show that the G-equivariant quantum cohomology QHG⁎(M) defines a canonical holomorphic Lagrangian subvariety LG(M)↪BFM(GC∨) in the BFM-space of the Langlands dual group, confirming an expectation of Teleman from [51].

Schubert Class and Cyclotomic NilHecke Algebras

Let [Formula: see text] and [Formula: see text] be positive integers such that [Formula: see text], and let [Formula: see text] be the Grassmannian which consists of the set of [Formula: see text]-dimensional subspaces of [Formula: see text]. There is a [Formula: see text]-graded algebra isomorphism between the cohomology [Formula: see text] of [Formula: see text] and a natural [Formula: see text]-form [Formula: see text] of the [Formula: see text]-graded basic algebra of the type [Formula: see text] cyclotomic nilHecke algebra [Formula: see text]. We show that the isomorphism can be chosen such that the image of each (geometrically defined) Schubert class [Formula: see text] coincides with the basis element [Formula: see text] constructed by Hu and Liang by purely algebraic method, where [Formula: see text] with [Formula: see text] for each [Formula: see text], and [Formula: see text] is the [Formula: see text]-multipartition of [Formula: see text] associated to [Formula: see text]. A similar correspondence between the Schubert class basis of the cohomology of the Grassmannian [Formula: see text] and the [Formula: see text]'s basis ([Formula: see text] is an [Formula: see text]-multipartition of [Formula: see text] with each component being either [Formula: see text] or empty) of the natural [Formula: see text]-form [Formula: see text] of the [Formula: see text]-graded basic algebra of [Formula: see text] is also obtained. As an application, we obtain a second version of the Giambelli formula for Schubert classes.

A DG-extension of symmetric functions arising from higher representation theory

We investigate analogs of symmetric functions arising from an extension of the nilHecke algebra defined by Naisse and Vaz. These extended symmetric functions form a subalgebra of the polynomial ring tensored with an exterior algebra. We define families of bases for this algebra and show that it admits a family of differentials making it a sub-DG-algebra of the extended nilHecke algebra. The ring of extended symmetric functions equipped with this differential is quasi-isomorphic to the cohomology of a Grassmannian. We also introduce new deformed differentials on the extended nilHecke algebra that when restricted makes extended symmetric functions quasi-isomorphic to $GL(N)$-equivariant cohomology of Grassmannians.

Extended nilHecke algebras and symmetric functions in type B

We formulate a type B extended nilHecke algebra, following the type A construction of Naisse and Vaz. We describe an action of this algebra on extended polynomials and describe some results on the structure on the extended symmetric polynomials. Finally, following Appel, Egilmez, Hogancamp, and Lauda, we prove a result analogous to a classical theorem of Solomon connecting the extended symmetric polynomial ring to a ring of usual symmetric polynomials and their differentials.

On 2-Verma modules for quantum $${\mathfrak {sl}_{2}}$$ sl 2

In this paper we study the superalgebra $$A_n$$ , introduced by the authors in previous work on categorification of Verma modules for quantum $${\mathfrak {sl}_{2}}$$ . The superalgebra $$A_n$$ is akin to the nilHecke algebra, and shares similar properties. We also prove a uniqueness result about 2-Verma modules on $$\Bbbk $$ -linear 2-categories.

Towards functor exponentiation

We consider a possible framework to categorify the exponential map exp⁡(−f) given the categorification of a generator f of sl2 by Lauda. In this setup the Taylor expansions of exp⁡(−f) and exp⁡(f) turn into complexes built out of categorified divided powers of f. Hom spaces between tensor powers of categorified f are given by diagrammatics combining nilHecke algebra relations with those for a additional “short strand” generator. The proposed framework is only an approximation to categorification of exponentiation, because the functors categorifying exp⁡(f) and exp⁡(−f) are not invertible.

Spin nilHecke algebras of classical type

We formulate and study the spin nilHecke algebras NHn–b and NHn–d 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 Poln– 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 Poln– 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.

Chern class of Schubert cells in the flag manifold and related algebras

We discuss a relationship between Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds, the Fomin–Kirillov algebra, and the generalized nil-Hecke algebra. We show that the nonnegativity conjecture in the Fomin–Kirillov algebra implies the nonnegativity of the Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds for type A. Motivated by this connection, we also prove that the (equivariant) Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds are certain summations of the structure constants of the equivariant cohomology of Bott–Samelson varieties. We also discuss refined positivity conjectures of the Chern–Schwartz–MacPherson classes for Schubert cells motivated by the nonnegativity conjecture in the Fomin–Kirillov algebra.

The Differential Graded Odd NilHecke Algebra

We equip the odd nilHecke algebra and its associated thick calculus category with digrammatically local differentials. The resulting differential graded Grothendieck groups are isomorphic to two different forms of the positive part of quantum sl(2) at a fourth root of unity.

Quantum Cohomology via Vicious and Osculating Walkers

We relate the counting of rational curves intersecting Schubert varieties of the Grassmannian to the counting of certain non-intersecting lattice paths on the cylinder, so-called vicious and osculating walkers. These lattice paths form exactly solvable statistical mechanics models and are obtained from solutions to the Yang–Baxter equation. The eigenvectors of the transfer matrices of these models yield the idempotents of the Verlinde algebra of the gauged \({\mathfrak{\hat{u}}(n)_{k}}\)-WZNW model. The latter is known to be closely related to the small quantum cohomology ring of the Grassmannian. We establish further that the partition functions of the vicious and osculating walker model are given in terms of Postnikov’s toric Schur functions and can be interpreted as generating functions for Gromov–Witten invariants. We reveal an underlying quantum group structure in terms of Yang–Baxter algebras and use it to give a generating formula for toric Schur functions in terms of divided difference operators which appear in known representations of the nil-Hecke algebra.

On the Topology of Kac–Moody groups

We study the topology of spaces related to Kac–Moody groups. Given a Kac–Moody group over $$\mathbb C $$ , let $$\text {K}$$ denote the unitary form with maximal torus $${{\mathrm{T}}}$$ having normalizer $${{\mathrm{N}}}({{\mathrm{T}}})$$ . In this article we study the cohomology of the flag manifold $$\text {K}/{{{\mathrm{T}}}}$$ as a module over the Nil-Hecke algebra, as well as the (co)homology of $$\text {K}$$ as a Hopf algebra. In particular, if $$\mathbb F $$ has positive characteristic, we show that $$\text {H}_*(\text {K},\mathbb F )$$ is a finitely generated algebra, and that $$\text {H}^*(\text {K},\mathbb F )$$ is finitely generated only if $$\text {K}$$ is a compact Lie group . We also study the stable homotopy type of the classifying space $$\text {BK}$$ and show that it is a retract of the classifying space $$\text {BN(T)}$$ of $${{\mathrm{N}}}({{\mathrm{T}}})$$ . We illustrate our results with the example of rank two Kac–Moody groups.

The Odd nilHecke Algebra and its Diagrammatics

We introduce an odd version of the nilHecke algebra and develop an odd analogue of the thick diagrammatic calculus for nilHecke algebras. We graphically describe idempotents which give a Morita equivalence between odd nilHecke algebras and the rings of odd symmetric functions in finitely many variables. Cyclotomic quotients of odd nilHecke algebras are Morita equivalent to rings which are odd analogues of the cohomology rings of Grassmannians. Like their even counterparts, odd nilHecke algebras categorify the positive half of quantum sl(2).

Affine Nil-Hecke Algebras and Braided Differential Structure on Affine Weyl Groups

We construct a model of the affine nil-Hecke algebra as a subalgebra of the Nichols-Woronowicz algebra associated to a Yetter-Drinfeld module over the affine Weyl group. We also discuss the Peterson isomorphism between the homology of the affine Grassmannian and the small quantum cohomology ring of the flag variety in terms of the braided differential calculus.