Alladi Ramakrishnan Hall

Eigenvalues and Eigenvectors of the perfect matching association scheme

Murali K. Srinivasan

IIT Bombay

We revisit the Bose-Mesner algebra of the perfect matching scheme or, equivalently, the Hecke algebra of the Gelfand pair $(S_2n, H_n)$ ($H_n$ = hyperoctahedral group). Our main results are:

(i) An algorithm for the eigenvalues from symmetric group characters by solving linear equations.

(ii) Universal formulas, as content evaluations of symmetric functions, for the eigenvalues of fixed orbitals (generalizing
a result of Diaconis and Holmes).

(iii) Inductive construction of the eigenvectors (generalizing a result of Godsil and Meagher).