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Advances in Combinatorial Mathematics - Proceedings of the Waterloo Workshop in Computer Algebra 2008 (Paperback, 2010 ed.)
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Advances in Combinatorial Mathematics - Proceedings of the Waterloo Workshop in Computer Algebra 2008 (Paperback, 2010 ed.)
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It is a pleasure for me to have the opportunity to write the
foreword to this volume, which is dedicated to Professor Georgy
Egorychev on the occasion of his seventieth birthday. I have
learned a great deal from his creative and important work, as has
the whole world of mathematics. From his life's work (so far) in
having made d- tinguished contributions to ?elds as diverse as the
theory of permanents, Lie groups, combinatorial identities, the
Jacobian conjecture, etc., let me comment on just two of the most
important of his research areas. The permanent of an nxn matrix A
is Per(A)= a a ...a , (1) ? 1,i 2,i n,i 1 2 n extended over the n!
permutations{i ,...,i} of{1,2,...,n}. Thus, the permanent 1 n is
"like the determinant except for dropping the sign factors from the
terms." H- ever by dropping those signs, one loses almost all of
the friendly characteristics of determinants, such as the fact that
det(AB)= det(A)det(B), the invariance under elementary row and
column operations, and so forth. The permanent is a creature of
multilinear algebra, rather than of linear algebra, and is much
crankier to deal with in virtually all of its aspects, both
theoretical and algorithmic.
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