Abstract
We revisit the Bose-Mesner algebra of the perfect matching association scheme. Our main results are: 1. An inductive algorithm, based on solving linear equations, to compute the eigenvalues of the orbital basis elements given the central characters of the symmetric groups. 2. Universal formulas, as content evaluations of symmetric functions, for the eigenvalues of fixed orbitals. 3. An inductive construction of an eigenvector (the so called first Gelfand-Tsetlin vector) in each eigenspace leading to a different inductive algorithm (not using central characters) for the eigenvalues of the orbital basis elements.
Highlights
In this paper we revisit the Bose–Mesner algebra of the perfect matching association scheme
The eigenspaces of B2n, in its left action on C[M2n], are indexed by even Young diagrams with 2n boxes (i.e. Young diagrams with 2n boxes having an even number of boxes in every row) and the orbital basis of B2n is indexed by even partitions of 2n
We show that we can inductively compute the eigenvalues of B2, B4, . . . , B2n from the central characters of S2, S4, . . . , S2n by solving systems of linear equations
Summary
In this paper we revisit the Bose–Mesner algebra of the perfect matching association scheme. Let N2μ denote the orbital basis element of B2n indexed by the even partition 2μ and let θ22μλ, λ, μ n, denote the eigenvalue (which can be shown to be an integer, see Section 2) of N2μ on V 2λ. Let Θ (2n) denote the eigenvalue table of B2n, i.e. Θ (2n) is the Yn × Pn matrix with entry in row λ, column μ given by θ22μλ. We call {Nμ | μ ∈ Pn} the orbital basis of Bn. Parts (ii) and (iv) of Lemma 2.1 show that the eigenvalues of Nμ are integers using the following standard argument (and the fact that the irreducible characters of Sn are integer valued). (i) Given the central characters, we shall simultaneously inductively calculate the eigenvalues and intersection numbers of B2n.
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