Representations of sl(2) in the Boolean lattice, and the Hamming and Johnson schemes

TL;DR

This paper explores the representation of the Lie algebra sl(2) within the Boolean lattice framework, revealing connections to special functions, Krawtchouk polynomials, and combinatorial schemes like Hamming and Johnson, with applications to Hadamard matrices.

Contribution

It introduces a novel Boolean lattice representation of sl(2) derived from zeon algebra, detailing its structure and connections to classical combinatorial schemes and special functions.

Findings

Decomposition of Boolean lattice representation into irreducible su(2) modules

Identification of group elements and special functions in this context

Applications to Boolean posets and Hadamard-Sylvester matrices

Abstract

Starting with the zero-square "zeon algebra", the regular representation gives rise to a Boolean lattice representation of sl(2). We detail the su(2) content of the Boolean lattice, providing the irreducible representations carried by the algebra generated by the subsets of an n-set. The group elements are found, exhibiting the "special functions" in this context. The corresponding Leibniz rule and group law are shown. Krawtchouk polynomials, the Hamming and the Johnson schemes appear naturally. Applications to the Boolean poset and the structure of Hadamard-Sylvester matrices are shown as well.