J Algebr Comb (2011) 33: 163–198 DOI 10.1007/s10801-010-0238-4 A computational and combinatorial exposé of plethystic calculus Nicholas A. Loehr · Jeffrey B. Remmel Received: 29 January 2010 / Accepted: 19 May 2010 / Published online: 12 June 2010 © The Author(s) 2010. This article is published with open access at Springerlink.com Abstract In recent years, plethystic calculus has emerged as a powerful technical tool for studying symmetric polynomials. In particular, some striking recent advances in the theory of Macdonald polynomials have relied heavily on plethystic computa- tions. The main purpose of this article is to give a detailed explanation of a method for finding combinatorial interpretations of many commonly occurring plethystic ex- pressions, which utilizes expansions in terms of quasisymmetric functions. To aid newcomers to plethysm, we also provide a self-contained exposition of the funda- mental computational rules underlying plethystic calculus. Although these rules are well-known, their proofs can be difficult to extract from the literature. Our treatment emphasizes concrete calculations and the central role played by evaluation homomor- phisms arising from the universal mapping property for polynomial rings. Keywords Plethysm · Symmetric functions · Quasisymmetric functions · LLT polynomials · Macdonald polynomials 1 Introduction 1.1 Plethysm The plethysm F [G] of a symmetric polynomial F(x) with a symmetric polynomial G(x) is essentially the polynomial obtained by substituting the monomials of G(x) First author supported in part by National Security Agency grant H98230-08-1-0045. N.A. Loehr () Virginia Tech, Blacksburg, VA 24061-0123, USA e-mail:
[email protected] J.B. Remmel University of California, San Diego, La Jolla CA 92093-0112, USA e-mail:
[email protected] 164 J Algebr Comb (2011) 33: 163–198 for the variables of F(x).