An elementary construction of the ring of dual $K$-$Q$-cancellation property

TL;DR

This paper offers a straightforward algebraic construction of dual $K$-$Q$-functions, simplifying previous approaches and demonstrating their generation by functions linked to odd row partitions.

Contribution

It introduces an elementary, purely algebraic construction of dual $K$-$Q$-functions, avoiding fermionic operators and vacuum expectations.

Findings

The algebra of dual $Q$-cancellation property is generated by dual $K$-$Q$-functions.

Provides a simpler, algebraic approach to $K$-theoretic $Q$-functions.

Shows the generation of the algebra by functions associated with odd row partitions.

Abstract

This paper presents an elementary introduction on $K$-theoretic $Q$-functions, which were introduced by Ikeda and Naruse in 2013. These functions, which serve as $K$-theoretic analogs of Schur $Q$-functions, are known to possess combinatorial and algebraic constructions. In a 2022 paper, the author introduced ``$\beta$-deformed power-sums" to provide a simpler, more algebraic construction of these functions. Since the original approach relies on fermionic operators and vacuum expectation values, this paper presents a more accessible, purely algebraic treatment, following the exposition of Schur $Q$-functions in Macdonald's standard textbook. We also show that the algebra of dual $Q$-cancellation property with integer coefficients is generated by dual $K$-$Q$-functions associated with an odd row partition.