A proof of the Göllnitz-Gordon-Andrews identities via commutative algebra

The Göllnitz-Gordon-Andrews identities generalize the partition identities discovered independently by H. Göllnitz and B. Gordon. In this article, we present a commutative algebra proof of the Göllnitz-Gordon-Andrews identities. More generally, we establish a family of identities, the special cases of which are the Göllnitz-Gordon-Andrews identities. In the proof, we relate the generating functions associated with these identities to the Hilbert-Poincaré series of suitably constructed graded algebras.

💡 Research Summary

The paper presents a new proof of the Göllnitz‑Gordon‑Andrews (G‑G‑A) partition identities using tools from commutative algebra. After recalling the classical partition identities of Euler, Rogers‑Ramanujan, Göllnitz, and Gordon, the authors introduce the G‑G‑A identities in their most general form: for integers (r\ge2) and (1\le i\le r), the number (C_{r,i}(n)) of partitions of (n) into parts avoiding certain residues modulo (4r) equals the number (D_{r,i}(n)) of partitions satisfying a set of difference and parity conditions. The case (r=2) recovers the original Göllnitz‑Gordon identities.

A more general family of generating functions is then defined. For a non‑negative integer (J) and (1\le i\le r), the series (C_{(r-1)J+\ell}(q)) (with (\ell=r-i+1)) and (E_{r,i,J}(q)) are introduced, where (E_{r,i,J}(n)) counts partitions with the same difference and parity constraints as (D_{r,i}(n)) but with all parts larger than (2J) and at most (i-1) parts equal to (2J+1) or (2J+2). Coulson et al. proved that (C_{(r-1)J+\ell}(q)=E_{r,i,J}(q)) for all (J\ge0); the case (J=0) yields the G‑G‑A identities, while for (J>0) a direct combinatorial interpretation was previously unknown.

The authors’ strategy is to reinterpret these generating functions as Hilbert‑Poincaré series of suitably constructed graded algebras. They work over a field (F) of characteristic zero and consider the infinite polynomial ring (S=F