GENERALIZED STRAIGHTENING LAWS FOR PRODUCTS OF DETERMINANTS by BRIAN

The (semi)standard Young tableau have been known since Hodge and Littlewood to naturally index a basis for the multihomogeneous coordinate rings of flag varieties under the Plicker embedding. In representation theory, the irreducible representations of GL,(C) arise as the multihomogeneous components of these rings. I introduce a new class of straight tableau by slightly weakening the requirements for standard tableaux. The straight tableau are defined for the more general class of row-convex shapes. I show for any row-convex shape that these tableaux index a basis for the associated representation. I provide an explicit straightening algorithm for expanding elements of the representation into this basis. For skew shapes, this algorithm specializes to the classical straightening law. I define the anti-straight tableaux, which provide a similar basis and straightening law for the kernels of the projection maps in certain James-Peel complexes. The above results allow me to provide degree 2 Groebner bases for the homogeneous coordinate rings of certain of the Magyar "configuration varieties." Further, I provide canonical subalgebra (or SAGBI) bases for these rings. I establish a quantum version of the basis results and straightening algorithms, using a new notion of SAGBI bases suitable for this non-commutative setting. A benefit of this technique is a new (and strengthened) proof of the Huang-Zhang standard basis theorem for quantum bitableaux. Rota and his collaborators have generalized the commutative constructions for GL(V) representations to the setting of superalgebras and representations of the general linear Lie superalgebras. The notions of straight and anti-straight tableaux are defined in this general context, thus allowing the case of Weyl and Schur modules to be handled concurrently. I show in this setting how the basis indexed by the straight tableaux is naturally related to a basis for the dual representation given by a supersymmetric version of the Reiner-Shimozono decomposable tableaux. Thesis Supervisor: Gian-Carlo Rota Title: Professor of Applied Mathematics Acknowledgments I am pleased to thank my advisor, Gian-Carlo Rota. He has been extremely generous personally as well as mathematically and his stunning courses shnrowed me that I truly wanted to be a mathematician. I am indebted to David Buchsbaum for numerous conversations and especially for pointing out that Woodcock's work pleads to be made combinatorial. I have been remarkably lucky to have been in the same locale as Peter Magyar and Mark Shimozono. I have profited greatly from them and my interactions with the latter provided much of the impetus for writing Chapter 4. I would also like to thank David Eisenbud for several helpful conversations about Groebner bases and Bernd Sturmfels for sending me his preprint [Stu94]. Mathematics is of course comprised of people as well as ideas. I owe much to the support of more mathematicians than I can readily list-thank you all. I owe especial thanks to Richard Ehrenborg, Gabor Hetyei, Lauren Rose and Marguerite EisensteinTaylor. The last has listened to me rehearse, and then ruthlessly improved, each of my talks. I would be remiss in not thanking Steve Maurer, my mentor at Swarthmore, for first showing me that mathematics was beautiful, that algorithms were part and parcel of it, and that one could be a good parent whilst being a mathematician. Everyone should have a life outside mathematics and even if I have sometimes put said life (and my former housemates) on hold, I am very grateful to Lori Kenschaft and Diana Stiefbold. My family already knows well how important they have been to me. I would like to thank my parents, Harris and Diana Taylor, for their constant love and support. I have spent many hours on the phone with family; thanks to my mom for allowing me to talk math with her and to my sister, Rebecca, for listening to me complain. My grandparents, Lawrence and Beatrice Kahn, provided me, as both an undergraduate and a graduate student, with financial support that was, if possible, more than consonant with the value they have placed on education. Such rare generosity is particularly valuable in the present times and is priceless for having come, no strings attached, from two such wonderful people. I wish my grandmother could have seen this work finished. But she saw it started, and she, especially, understood that learning is an ongoing process. Finally, I cannot here properly thank my wife, Marguerite; she has had to bear the brunt of my unhappiness and frustration in graduate school while being a graduate student herself. To say that her love, support, and companionship have been invaluable is to vastly understate the truth. I am pleased to thank the Office of Naval Research for the support of an ONRNDSEG fellowship.