Pfaffian and Hafnian identities in shuffle algebras. (English) Zbl 1018.05006

Luque, Jean-Gabriel; Thibon, Jean-Yves

Adv. Appl. Math. 29, No. 4, 620-646 (2002).

Let \(\mathbb{Z}\langle A\rangle\) be the free associative ring over an alphabet \(A\) with comultiplication which admits the Lie polynomials as primitive elements. The shuffle product is an operation on \(\mathbb{Z}\langle A\rangle\) which is dual to this comultiplication. For a vector space \(\mathcal H\) of integrable functions over an interval \((a,b)\) K.-T. Chen [Proc. Lond. Math. Soc., III. Ser. 4, 502-512 (1954; Zbl 0058.25603) and Ann. Math. (2) 65, 163-178 (1957; Zbl 0077.25301] related the linear form on \({\mathcal H}^{\otimes n}\) \[ \langle f_1\otimes\cdots\otimes f_n\rangle =\int_a^b dx_1\int_a^{x_1} dx_2\;\cdots \int_a^{x_{n-1}} dx_nf_1(x_1)f_2(x_2)\cdots f_n(x_n) \] and expressed the product \(\langle u\rangle\langle v\rangle\) in terms of the shuffle product. This implies that certain identities involving multiple integrals, such as the de Bruijn and Wick formulas, amount to combinatorial identities for Pfaffians and Hafnians in shuffle algebras. In the paper under review the authors provide direct algebraic proofs of such shuffle identities, and obtain various generalizations. They also discuss some Pfaffian identities due to Sundquist and Ishiwara-Wakayama, and a Cauchy formula for anticommutative symmetric functions. They also extend some of the previous considerations to hyper-Pfaffians and hyper-Hafnians.

Reviewer: Vesselin Drensky (Sofia)

MSC:

05A19	Combinatorial identities, bijective combinatorics
05E05	Symmetric functions and generalizations
16S10	Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.)
16W99	Associative rings and algebras with additional structure

Keywords:

shuffle algebras; Pfaffian identities; Hafnian identities; de Bruijn formula; Wick formula; multiple integrals; combinatorial identities; shuffle identities; symmetric functions

Citations:

Zbl 0058.25603; Zbl 0077.25301

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