✦ LIBER ✦

Bijective matrix algebra

✍ Scribed by Nicholas A. Loehr; Anthony Mendes

Publisher: Elsevier Science
Year: 2006
Tongue: English
Weight: 277 KB
Volume: 416
Category: Article
ISSN: 0024-3795
DOI: 10.1016/j.laa.2006.01.004

No coin nor oath required. For personal study only.

✦ Synopsis

If A and B are square matrices such that AB = I , then BA = I automatically follows. We prove a combinatorial version of this result in the case where the entries of A and B count collections of signed, weighted objects. Specifically, we give an algorithm that transforms any given bijective proof of the identity AB = I into an explicit bijective proof of the identity BA = I . Letting A and B be the Kostka matrix and its inverse, this settles an open problem posed by Egecioglu and Remmel in 1990.

📜 SIMILAR VOLUMES

Geometric matrix algebra

✍ Garret Sobczyk 📂 Article 📅 2008 🏛 Elsevier Science 🌐 English ⚖ 140 KB

Algebra of Matrix Arithmetic

✍ Gautami Bhowmik; Olivier Ramaré 📂 Article 📅 1998 🏛 Elsevier Science 🌐 English ⚖ 207 KB

We study the algebra of the arithmetic of integer matrices. A link is established between the divisor classes of matrices and lattices. The algebra of arithmetical functions of integral matrices is then shown to be isomorphic to an extension of the Hecke algebra, also called a Hall algebra in combin

Three Alternating Sign Matrix Identities

Three Alternating Sign Matrix Identities in Search of Bijective Proofs

✍ David M. Bressoud 📂 Article 📅 2001 🏛 Elsevier Science 🌐 English ⚖ 121 KB

This paper highlights three known identities, each of which involves sums over alternating sign matrices. While proofs of all three are known, the only known derivations are as corollaries of difficult results. The simplicity and natural combinatorial interpretation of these identities, however, sug

Isometries of matrix algebras

✍ Robert Grone; Marvin Marcus 📂 Article 📅 1977 🏛 Elsevier Science 🌐 English ⚖ 412 KB

A combinatorial approach to matrix algeb

A combinatorial approach to matrix algebra

✍ Doron Zeilberger 📂 Article 📅 1985 🏛 Elsevier Science 🌐 English ⚖ 480 KB

The theory of correspondence reaches far deeper than that of mere numerical congruity with which it is associated as the substance with the shadow"

Friedland–Hersonsky problem for matrix a

Friedland–Hersonsky problem for matrix algebra

✍ Wensheng Cao; Xiantao Wang 📂 Article 📅 2003 🏛 Elsevier Science 🌐 English ⚖ 119 KB

In this paper we show that the answer to problem 3.9 in [Duke Math.