Projective geometries, $Q$-polynomial structures, and quantum groups

Abstract

In 2023 we obtained a $Q$-polynomial structure for the projective geometry $L_N(q)$. In the present paper, we display a more general $Q$-polynomial structure for $L_N(q)$. Our new $Q$-polynomial structure is defined using a free parameter $\phi$ that takes any positive real value. For $\phi =1$ we recover the original $Q$-polynomial structure. We interpret the new $Q$-polynomial structure using the quantum group $U_{q^{1/2}}({\mathfrak{sl}}_2)$ in the equitable presentation. We use the new $Q$-polynomial structure to obtain analogs of the four split decompositions that appear in the theory of $Q$-polynomial distance-regular graphs.

30 pages

Cited by in corpus (1)

• The nucleus of a $Q$-polynomial distance-regular graph