A bijective proof of the second reduction formula for Littlewood-Richardson coefficients

Bull. Korean Math. Soc. 2008 Vol. 45, No. 3, 485-494 Printed September 1, 2008

Soojin Cho, Eun-Kyoung Jung, and Dongho Moon Ajou University, Ajou University, Sejong University

Abstract : There are two well known reduction formulae for structural constants of the cohomology ring of Grassmannians, i.e., Littlewood-Richardson coefficients. Two reduction formulae are a conjugate pair in the sense that indexing partitions of one formula are conjugate to those of the other formula. A nice bijective proof of the first reduction formula is given in the authors' previous paper while a (combinatorial) proof for the second reduction formula in the paper depends on the identity between Littlewood-Richardson coefficients of conjugate shape. In this article, a $direct$ bijective proof for the second reduction formula for Littlewood-Richardson coefficients is given. Our proof is independent of any previously known results (or bijections) on tableaux theory and supplements the arguments on bijective proofs of reduction formulae in the authors' previous paper.