The Odd nilHecke Algebra and its Diagrammatics

Ellis, Alexander P.; Khovanov, Mikhail; Lauda, Aaron D.

doi:10.1093/imrn/rns240

Cited by 40 publications

(89 citation statements)

References 12 publications

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“…As a byproduct of this investigation, an "odd" analogue O of the algebra was defined and found to admit signed analogues of many of the combinatorial properties of [3,4]. We gave two different candidates for elements playing the role analogous to that of Schur functions.…”

Section: Outline Of This Papermentioning

confidence: 99%

The odd Littlewood–Richardson rule

Ellis

2012

J Algebr Comb

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In previous work with Mikhail Khovanov and Aaron Lauda we introduced two odd analogues of the Schur functions: one via the combinatorics of Young tableaux (odd Kostka numbers) and one via an odd symmetrization operator. In this paper we introduce a third analogue, the plactic Schur functions. We show they coincide with both previously defined types of Schur function, confirming a conjecture. Using the plactic definition, we establish an odd Littlewood-Richardson rule. We also re-cast this rule in the language of polytopes, via the Knutson-Tao hive model.

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Section: Outline Of This Papermentioning

confidence: 99%

The odd Littlewood–Richardson rule

Ellis

2012

J Algebr Comb

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“…It is known that the even Khovanov homology can be recovered from categorical representations of categorified U q (sl 2 ) [We10]. A recent discovery of odd nilHecke algebras [EKL12,KKT11], which categorifies the negative half of U q (sl 2 ), suggests the existence of the odd Khovanov homology may also possess a representation-theoretical explanation.…”

Section: Introductionmentioning

confidence: 99%

A 2-category of chronological cobordisms and odd Khovanov homology

Putyra

2014

Banach Center Publ.

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Abstract. We create a framework for odd Khovanov homology in the spirit of Bar-Natan's construction for the ordinary Khovanov homology. Namely, we express the cube of resolutions of a link diagram as a diagram in a certain 2-category of chronological cobordisms and show that it is 2-commutative: the composition of 2-morphisms along any 3-dimensional subcube is trivial. This allows us to create a chain complex, whose homotopy type modulo certain relations is a link invariant. Both the original and the odd Khovanov homology can be recovered from this construction by applying certain strict 2-functors. We describe other possible choices of functors, including the one that covers both homology theories and another generalizing dotted cobordisms to the odd setting. Our construction works as well for tangles and is conjectured to be functorial up to sign with respect to tangle cobordisms.

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“…The odd nilHecke algebras [19] and quiver Hecke(-Clifford) superalgebras of [22] should satisfy the (super analogue of) the axioms of B-quasihereditary for a class B of algebras which are built out of polynomial and Clifford algebras.…”

Section: 4mentioning

confidence: 99%

Affine highest weight categories and affine quasihereditary algebras

Kleshchev

2015

Proceedings of the London Mathematical Society

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Abstract. Koenig and Xi introduced affine cellular algebras. Kleshchev and Loubert showed that an important class of infinite dimensional algebras, the KLR algebras R(Γ) of finite Lie type Γ, are (graded) affine cellular; in fact, the corresponding affine cell ideals are idempotent. This additional property is reminiscent of the properties of quasihereditary algebras of Cline-Parshall-Scott in a finite dimensional situation. A fundamental result of Cline-Parshall-Scott says that a finite dimensional algebra A is quasihereditary if and only if the category of finite dimensional A-modules is a highest weight category. On the other hand, S. Kato and Brundan-Kleshchev-McNamara proved that the category of finitely generated graded R(Γ)-modules has many features reminiscent of those of a highest weight category. The goal of this paper is to axiomatize and study the notions of an affine quasihereditary algebra and an affine highest weight category. In particular, we prove an affine analogue of the Cline-Parshall-Scott Theorem. We also develop stratified versions of these notions.

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The Odd nilHecke Algebra and its Diagrammatics

Cited by 40 publications

References 12 publications

The odd Littlewood–Richardson rule

The odd Littlewood–Richardson rule

A 2-category of chronological cobordisms and odd Khovanov homology

Affine highest weight categories and affine quasihereditary algebras

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