2025/08/15 by Laura Pierson, Pierson, Laura
Engineering · #Combinatorics (math.CO) #FOS: Mathematics #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2508.11813
openalex publication_date 2025/08/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Several families of polynomials of combinatorial and representation theoretic interest (notably the Schur polynomials sλ, Demazure characters \mathfrakDa, and Demazure atoms \mathfrakAa) can be defined in terms of divided difference operators. Hivert (2000) defines "fundamental analogues" of these divided difference operators, and Hivert and Hicks-Niese show in arXiv:2406.02420 that the polynomials that arise from those fundamental operators in analogous ways to the three families of polynomials above are respectively the fundamental quasisymmetric functions Fa from (1984), the fundamental slides \mathfrakFa of Assaf and Searles from arXiv:1603.09744, and the fundamental particles \mathfrakPa of Searles from arXiv:1707.01172. Lascoux (2001) defines K-analogues of the divided difference operators, and in arXiv:1908.07364, Buciumas, Scrimshaw, and Weber show that the polynomials arising in corresponding ways from the K-theoretic divided difference operators are respectively the Grothendieck polynomials sλ, the combinatorial Lascoux polynomials \mathfrakDa from arXiv:1611.08777, and the combinatorial Lascoux atoms \mathfrakAa from arXiv:1611.08777, as conjectured by Monical in arXiv:1611.08777. We define K-analogues of Hivert's fundamental divided difference operators and show that the polynomials arising in the corresponding ways from our new operators are respectively the multifundamentals Fa of Lam and Pylyavskyy from arXiv:0705.2189, the fundamental glides \mathfrakFa from of Pechenik and Searles from arXiv:1611.02545, and the kaons \mathfrakPa of Monical, Pechenik, and Searles from arXiv:1806.03802.