P-Partitions and a Multi-Parameter Klyachko Idempotent

Because they play a role in our understanding of the symmetric group algebra, Lie idempotents have received considerable attention. The Klyachko idempotent has attracted interest from combinatorialists, partly because its denition involves the major index of permutations. For the symmetric group Sn, we look at the symmetric group algebra with coecients from the eld of rational functions in n variables q1;:::;qn .I n this setting, we can dene an n-parameter generalization of the Klyachko idempotent, and we show it is a Lie idempotent in the appropriate sense. Somewhat surprisingly, our proof that it is a Lie element emerges from Stanley’s theory of P -partitions.