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Combinatorics of $$K$$-theory via a $$K$$-theoretic Poirier-Reutenauer bialgebra. (English) Zbl 1328.05193
Summary: We use the $$K$$-Knuth equivalence of A. Buch and M. Samuel [“$$K$$-theory of minuscule varieties”, J. Reine Angew. Math. (to appear)] to define a $$K$$-theoretic analogue of the Poirier-Reutenauer Hopf algebra. As an application, we rederive the $$K$$-theoretic Littlewood-Richardson rules of H. Thomas and A. Yong [Algebra Number Theory 3, No. 2, 121–148 (2009; Zbl 1229.05285); Int. Math. Res. Not. 2011, No. 12, 2766–2793 (2011; Zbl 1231.05280)] and of Buch and Samuel [loc. cit.].

##### MSC:
 05E05 Symmetric functions and generalizations 05A17 Combinatorial aspects of partitions of integers 14N15 Classical problems, Schubert calculus
##### Keywords:
bialgebra; $$K$$-theory; symmetric functions
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##### References:
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