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Results and conjectures on simultaneous core partitions
, 2014
"... An ncore partition is an integer partition whose Young diagram contains no hook lengths equal to n. We consider partitions that are simultaneously acore and bcore for two relatively prime integers a and b. These are related to abacus diagrams and the combinatorics of the affine symmetric group ..."
Abstract

Cited by 14 (2 self)
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An ncore partition is an integer partition whose Young diagram contains no hook lengths equal to n. We consider partitions that are simultaneously acore and bcore for two relatively prime integers a and b. These are related to abacus diagrams and the combinatorics of the affine symmetric group (type A). We observe that selfconjugate simultaneous core partitions correspond to the combinatorics of type C, and use abacus diagrams to unite the discussion of these two sets of objects. In particular, we prove that 2n and (2mn+1)core partitions correspond naturally to dominant alcoves in themShi arrangement of typeCn, generalizing a result of Fishel–Vazirani for typeA. We also introduce a major index statistic on simultaneous n and (n + 1)core partitions and on selfconjugate simultaneous 2n and (2n + 1)core partitions that yield qanalogues of the CoxeterCatalan numbers of type A and type C. We present related conjectures and open questions on the average size of a simultaneous core partition, qanalogs of generalized Catalan numbers, and generalizations to other Coxeter groups. We also discuss connections with the cyclic sieving phenomenon and q, tCatalan numbers.
BIJECTIVE PROJECTIONS ON PARABOLIC QUOTIENTS OF AFFINE WEYL GROUPS
"... Abstract. Affine Weyl groups and their parabolic quotients are used extensively as indexing sets for objects in combinatorics, representation theory, algebraic geometry, and number theory. Moreover, in the classical Lie types we can conveniently realize the elements of these quotients via intuitive ..."
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Abstract. Affine Weyl groups and their parabolic quotients are used extensively as indexing sets for objects in combinatorics, representation theory, algebraic geometry, and number theory. Moreover, in the classical Lie types we can conveniently realize the elements of these quotients via intuitive geometric and combinatorial models such as abaci, alcoves, coroot lattice points, core partitions, and bounded partitions. In [1] Berg, Jones, and Vazirani described a bijection between ncores with first part equal to k and (n − 1)cores with first part less than or equal to k, and they interpret this bijection in terms of these other combinatorial models for the quotient of the affine symmetric group by the finite symmetric group. In this paper we discuss how to generalize the bijection of BergJonesVazirani to parabolic quotients of affine Weyl groups in type C. We develop techniques using the associated affine hyperplane arrangement to interpret this bijection geometrically as a projection of alcoves onto the hyperplane containing their coroot lattice points. We are thereby able to analyze this bijective projection in the language of various additional combinatorial models developed by Hanusa and Jones in [10], such as abaci, core partitions, and canonical reduced expressions in the Coxeter group. 1.