CU Algebraic Lie Theory Seminar

CU Algebraic Lie Theory Seminar
Tropical Decomposition of Young's Partition Lattice Vivek Dhand, Michigan State
December 11, 2012 Math 350 2-3pm	Abstract Young's partition lattice L(m,n) consists of unordered partitions having m parts where each part is at most n, ordered by inclusion of the corresponding Young diagrams. Many years ago, Richard Stanley conjectured that this poset has a decomposition into saturated rank-symmetric chains. Despite many efforts, this conjecture has only been proved when min(m,n) is less than or equal to 4. In this talk, we decompose L(m,n) into level sets for certain tropical polynomials derived from the secant varieties of the rational normal curve in projective space, and we show that resulting subposets have an elementary raising and lowering algorithm. As a corollary, we obtain a symmetric chain decomposition for the subposet of L(m,n) consisting of "sufficiently generic" partitions.

Tropical Decomposition of Young's Partition Lattice

Vivek Dhand, Michigan State

December 11, 2012

Math 350

2-3pm

Abstract

Young's partition lattice L(m,n) consists of unordered partitions having m parts where each part is at most n, ordered by inclusion of the corresponding Young diagrams. Many years ago, Richard Stanley conjectured that this poset has a decomposition into saturated rank-symmetric chains. Despite many efforts, this conjecture has only been proved when min(m,n) is less than or equal to 4. In this talk, we decompose L(m,n) into level sets for certain tropical polynomials derived from the secant varieties of the rational normal curve in projective space, and we show that resulting subposets have an elementary raising and lowering algorithm. As a corollary, we obtain a symmetric chain decomposition for the subposet of L(m,n) consisting of "sufficiently generic" partitions.