OTTAWA, Ontario — The CMS is pleased to announce that Hugh Thomas and Alexander Yong are the recipients of the 2011 G. de B. Robinson Award for their paper “Multiplicity-Free Schubert Calculus”, published in the Canadian Mathematical Bulletin (53:1 2010, 171-186; http://dx.doi.org/10.4153/CMB-2010-032-x).
The G. de B. Robinson Award was inaugurated to recognize the publication of excellent papers in the Canadian Journal of Mathematics and the Canadian Mathematical Bulletin and to encourage the submission of the highest quality papers to these journals. The first award was presented in 1996. The 2011 award was selected from papers that appeared in the Canadian Mathematical Bulletin in 2009 and 2010.
Grassmannians are fundamental objects in algebraic geometry and topology, and they play roles in representation theory, combinatorics, and some applications of mathematics. An old result of Schubert is that any subvariety of a Grassmannian is homologous to a unique nonnegative integer linear combination of classes of Schubert varieties, which are the simplest subvarieties in a Grassmannian.
There are two natural candidates for the next simplest type of subvarieties. Multiplicity-free subvarieties are those whose linear combination has coefficients 0 or 1. A second are Richardson varieties, which are the intersection of two Schubert varieties, and whose linear combinations are related to tensor product decompositions in representation theory. In "Multiplicity-free Schubert Calculus", Thomas and Yong give a classification of multiplicity-free Richardson varieties in a Grassmannian, answering a question of Fulton.
Previously, Stembridge solved the representation theoretic-analog: classify multiplicity-free decompositions of tensor products of irreducible $gl_n$-representations. While related, it does not directly apply to Fulton's question. Thomas and Yong first employ a simple reduction to certain basic Richardson varieties, and they show that basic Richardson varieties are multiplicity-free if and only if they fall into Stembridge's classification.
The relation to representation theory and Stembridge's work establishes one direction. For the other direction, given a basic Richardson variety not in Stembridge's classification, they give a different reduction which decreases the coefficients, and then use combinatorics to show that some coefficients exceed 1. The beauty of this result is the elegant application of these two simple reductions.
Hugh Thomas was born and raised in Winnipeg. He did his undergraduate work at the University of Toronto, and his Ph.D. at the University of Chicago, under the direction of William Fulton. In 2004, after holding postdoctoral fellowships at the University of Western Ontario and the Fields Institute, he joined the faculty of the University of New Brunswick. He is interested in a variety of topics from algebra and combinatorics, including cluster algebras, representation theory of hereditary algebras, and Schubert calculus.
Alexander Yong attended the University of Waterloo, receiving a B.Math degree in 1998 and an M.Math in 1999. He obtained a doctorate from the University of Michigan in 2003, under the direction of Sergey Fomin. He held postdoctoral positions at the University of California, Berkeley, the Fields Institute and the University of Minnesota. He joined the University of Illinois at Urbana-Champaign in 2008 where he is presently an Assistant Professor in the Department of Mathematics. His research is in algebraic combinatorics.
For more information, contact:
Prof. Ken Davidson
Chair, CMS Publications Committee
Department of Pure Mathematics
University of Waterloo
Communications and Special Projects Officer
Canadian Mathematical Society
(613) 733-2662 ext. 728
About the Canadian Mathematical Society
The Canadian Mathematical Society (CMS) is the main national organization whose goal is to promote and advance the discovery, learning, and application of mathematics. The Society's activities cover the whole spectrum of mathematics including: scientific meetings, research publications, and the promotion of excellence in mathematics education at all levels. The CMS has a number of awards and prizes that are annually given to individuals in recognition of outstanding contributions to the advancement of mathematics.