# ODDIFICATION OF THE COHOMOLOGY OF TYPE A SPRINGER VARIETIES

@article{Lauda2012ODDIFICATIONOT, title={ODDIFICATION OF THE COHOMOLOGY OF TYPE A SPRINGER VARIETIES}, author={Aaron D. Lauda and Heather M. Russell}, journal={International Mathematics Research Notices}, year={2012}, volume={2014}, pages={4822-4854} }

We identify the ring of odd symmetric functions introduced by Ellis and Khovanov as the space of skew polynomials fixed by a natural action of the Hecke algebra at q = 1. This allows us to define graded modules over the Hecke algebra at q = 1 that are 'odd' analogs of the cohomology of type A Springer varieties. The graded module associated to the full flag variety corresponds to the quotient of the skew polynomial ring by the left ideal of nonconstant odd symmetric functions. The top…

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## References

SHOWING 1-10 OF 48 REFERENCES

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

- MathematicsElectron. J. Comb.
- 2010

For the class of regular nilpotent Hessenberg varieties, this paper conjecture a quotient presentation for the cohomology ring and exhibit an explicit basis for this quotient, and Tantalizing new evidence supports the conjecture for a subclass of regularnilpotent varieties called Peterson varieties.

### CROSSINGLESS MATCHINGS AND THE COHOMOLOGY OF (n, n) SPRINGER VARIETIES

- Mathematics
- 2002

In an earlier paper, the author introduced a collection of rings that control a categorification of the quantum sl(2) invariant of tangles. We prove that centers of these rings are isomorphics to the…

### Quantum quasi-symmetric functions and Hecke algebras

- Mathematics
- 1996

The algebra of quasi-symmetric functions is known to describe the characters of the Hecke algebra of type at v = 0. We present a quantization of this algebra, defined in terms of filtrations of…

### Hecke Algebras of TypeAwithq=−1

- Mathematics
- 1996

Abstract In this paper we study the decomposition matrices of the Hecke algebras of type A with q =−1 over a field of characteristic 0. We give explicit formulae for the columns of the decomposition…

### A categorification of finite-dimensional irreducible representations of quantum sl(2) and their tensor products

- Mathematics
- 2005

The purpose of this paper is to study categorifications of tensor products of finite dimensional modules for the quantum group for sl(2). The main categorification is obtained using certain…

### Hecke Algebra Actions on the Coinvariant Algebra

- Mathematics
- 1999

Abstract Two actions of the Hecke algebra of type A on the corresponding polynomial ring are studied. Both are deformations of the natural action of the symmetric group on polynomials, and keep…

### Skew shape representations are irreducible

- Mathematics
- 2004

In this paper all of the classical constructions of A. Young are generalized to affine Hecke algebras of type A. It is proved that the calibrated irreducible representations of the affine Hecke…

### Odd Khovanov homology

- Mathematics
- 2013

We describe an invariant of links in the three-sphere which is closely related to Khovanov's Jones polynomial homology. Our construction replaces the symmetric algebra appearing in Khovanov's…

### Double affine Hecke algebras for the spin symmetric group

- Mathematics
- 2006

We introduce a new class (in two versions) of rational double affine Hecke algebras (DaHa) associated to the spin symmetric group. We establish the basic properties of the algebras, such as PBW and…