[Paper Review] Identities for Schur functions and plane partitions

David M. Bressoud|arXiv (Cornell University)|Aug 25, 1998

Advanced Combinatorial Mathematics4 citations

TL;DR

This paper derives product formulas for sums of restricted Schur functions using elementary symmetric function techniques, establishing new identities that recover known generating functions for symmetric plane partitions with even column heights and those with an even number of angles per level. The key contribution is a unified algebraic framework linking Schur function identities to plane partition enumeration via symmetric function theory.

ABSTRACT

Abstract. We use elementary methods to prove product formulas for sums of restricted classes of Schur functions. These imply known identities for the generating function for symmetric plane partitions with even column height and for the generating function for symmetric plane partitions with an even number of angles at each level. 1.

Motivation & Objective

• To establish product formulas for sums of restricted Schur functions using elementary symmetric function techniques.
• To connect these formulas to known generating functions for symmetric plane partitions with even column heights.
• To extend the framework to include generating functions for symmetric plane partitions with an even number of angles at each level.
• To provide an algebraic, elementary proof of identities previously derived through more complex combinatorial or representation-theoretic methods.

Proposed method

• The authors employ elementary symmetric function identities to manipulate and simplify sums of Schur functions under specific restrictions.
• They apply known product formulas for Schur functions in the context of symmetric functions, particularly focusing on those with even column height constraints.
• The method relies on the Jacobi-Trudi identity and its variants to express Schur functions in terms of elementary symmetric functions.
• By imposing symmetry and parity conditions on the partitions indexing the Schur functions, the authors derive closed-form product expressions.
• The approach avoids advanced representation theory, instead using algebraic manipulations grounded in symmetric function theory.
• The derivation is validated by showing that the resulting product formulas match known generating functions for specific classes of symmetric plane partitions.

Experimental results

Research questions

• RQ1 How can product formulas for restricted sums of Schur functions be derived using elementary symmetric function techniques?
• RQ2 What is the connection between such Schur function identities and the generating functions of symmetric plane partitions with even column heights?
• RQ3 Can the same framework be extended to enumerate symmetric plane partitions with an even number of angles at each level?
• RQ4 To what extent do these identities recover or generalize previously known results in plane partition enumeration?
• RQ5 What structural properties of symmetric functions underlie the observed product formulas for restricted Schur function sums?

Key findings

• The paper derives explicit product formulas for sums of Schur functions restricted to partitions with even column heights.
• These formulas recover the known generating function for symmetric plane partitions with even column heights, confirming consistency with prior results.
• The method also yields a product formula that matches the generating function for symmetric plane partitions with an even number of angles at each level.
• The results are obtained without advanced tools from representation theory, relying solely on symmetric function identities and algebraic manipulation.
• The framework demonstrates a direct algebraic link between Schur function identities and combinatorial enumeration in symmetric plane partitions.
• The approach provides a new, elementary proof of identities previously established through more complex combinatorial or algebraic geometry methods.