The odd Littlewood–Richardson rule

Abstract

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.

Oddification of the Cohomology of Type A Springer Varieties

“…Odd symmetric functions also have natural bases corresponding to complete, monomial, and forgotten symmetric functions, as well as a Schur polynomials basis [13,14]. There is also an odd analog of the Littlewood-Richardson rule for Schur polynomials developed by Ellis [12]. Recall that the odd Schubert polynomials defined above are a homogeneous basis for OPol n as a free left and right OΛ nmodule of rank n!…”

An odd categorification of $U_q (\mathfrak{sl}_2)$

“…The odd nilHecke algebra has led to a number of surprising new structures including an odd analogue of the ring of symmetric functions [26][27][28] and odd analogues of the cohomology groups of Grassmannians [28] and of Springer varieties [52]. These structures possess combinatorics quite similar to those of their even counterparts.…”

On q-symmetric functions and q-quasisymmetric functions

“…Motivated by the "oddification" of Khovanov homology, Ellis, Khovanov and Lauda introduced the ring of odd symmetric functions and defined three versions of odd Schur functions in a sequence of papers [5], [6], [7]. In the last one, Ellis proved that all three definitions coincide to obtain the odd Littlewood-Richardson rule.…”

“…where oc ν λµ := S ∈SYT(ν/λ) rect(S )=Tµ (−1) inv((T λ ) S ) if λ ⊂ ν and 0 otherwise, called the odd Littlewood-Richardson number. Note that we haven't used the Littlewood-Richardson skew tableaux to characterize oc ν λµ as in [7]. In particular, the formula (5.4) implies that s…”

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In this paper, we construct the q-analogue of Poirier-Reutenauer algebras, related deeply with other q-combinatorial Hopf algebras. As an application, we use them to realize the odd Schur functions defined in [5], and naturally obtain the odd Littlewood-Richardson rule concerned in [7]. Moreover, we construct the refinement of the odd Schur functions, called odd quasisymmetric Schur functions, parallel to the consideration in [13].

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