Steven Glenn Jackson

I am an associate professor in the Mathematics Department of the University of Massachusetts in Boston, and hold a Ph.D. in mathematics from Yale University. I am also a member of the American Mathematical Society, the Mathematical Association of America, and the Mathematicians and Education Reform Forum, and a National Project NeXT Fellow.

Honors and Awards:

• Dolciani-Halloran National Project NExT Fellow, 2003--2004.
• Award for Outstanding Achievement in Research and Scholarship, UMB College of Science and Mathematics, 2004--2005.
• Visiting Scholar, National University of Singapore, January 2006.
• Award for Outstanding Overall Achievement, UMB College of Science and Mathematics, 2006--2007.

Research:

My research interests include invariant theory and representation theory. I am particularly interested in toric degenerations of categorical quotients of flag varieties and their use in the calculation of branching multiplicities.  Recently I have also been working with Alfred Noël on some problems connected with prehomogeneous spaces of Dynkin-Kostant type arising in the theory of nilpotent orbits in real reductive Lie algebras, and on some problems in the representation theory of Coxeter groups connected with the Atlas of Lie Groups and Representations Project.  In addition to this, I am investigating the structure of centralizers of maximal compact subgroups in the universal enveloping algebras associated with real reductive Lie groups.

Teacher Preparation and Education Policy:

Publications:

• Prehomogeneous spaces associated with complex nilpotent orbits, with Alfred Noël, Journal of Algebra 289 (2005), pp. 515--557.  A computation of isotypic decompositions and relative invariants of prehomogeneous spaces of Dynkin-Kostant type arising from conjugacy classes of nilpotent elements in complex reductive Lie algebras.
• A LiE subroutine for computing prehomogeneous spaces associated with complex nilpotent orbits, with Alfred Noël, Lecture Notes in Computer Science 3516 (2005), pp. 611--618.  An algorithm for determination of highest weights in prehomogeneous spaces of Dynkin-Kostant type, implemented in the computer algebra system LiE.
• A LiE subroutine for computing prehomogeneous spaces associated with real nilpotent orbits, with Alfred Noël, Lecture Notes in Computer Science 3482 (2005), pp. 512--521.  An algorithm for determination of highest weights in prehomogeneous spaces arising via the Kostant-Sekiguchi correspondence from nilpotent orbits in simple real Lie algebras of inner type.
• Polarizable theta-stable parabolic subalgebras and KC -saturation in the non-compact real forms of G2 and F4, with Alfred Noël, Lecture Notes in Computer Science 3992 (2006), pp. 422--429.  An algorithm for computation of the KC-saturation of the nilradical of a polarizable theta-stable parabolic.
• Prehomogeneous spaces associated with real nilpotent orbits, with Alfred Noël, Journal of Algebra 305 (2006), pp. 194--269.  An extension of our calculation of isotypic decompositions and relative invariants to prehomogeneous spaces arising via the Kostant-Sekiguchi correspondence from nilpotent orbits in real reductive Lie algebras.
• Prehomogeneous spaces associated with nilpotent orbits in simple real Lie algebras E₆₍₆₎ and E_6(-26) , with Alfred Noël, Experimental Mathematics 15 (2006), pp. 455--469.  Algorithmic determination of isotypic decompositions for exceptional simple real Lie algebras of non-inner type.
• Toric degeneration of branching algebras, with Roger Howe, Soo-Teck Lee, Eng-Chye Tan, and Jeb Willenbring, Advances in Mathematics 220 (2009), pp. 1809--1841.  Construction of canonical bases for symmetric pair branching algebras, leading to equivariant degeneration of branching varieties to affine toric varieties.
• A new approach to computing generators for U(g)^K, with Alfred Noël. An algorithm for computing generators for the centralizer of a maximal compact subgroup in the universal enveloping algebra of a reductive Lie algebra (to appear in Journal of Algebra).
• A short note on nilpotent orbits associated to Coxeter cells, with Alfred Noël (to appear in ACM Communications in Computer Algebra).
  
• A geometric proof of the Newell-Littlewood rule.  A proof that certain products of classical nullcones admit flat equivariant deformations to products of a single nullcone and an affine space, leading to simple proof of the Newell-Littlewood rule governing decomposition of the corresponding classical tensor products (submitted).
• Nilpotent orbits associated to Coxeter cells, with Alfred Noël. An algorithm to identify irreducible representations of  classical Weyl groups from their tau-signatures (i.e. the sets of parabolic subgroups admitting sign characters in the restriction). For cell representations (and hence for their special irreducible components) the tau-signature can be read off immediately from the W-graph; in particular, in the context of the Atlas of Lie Groups and Representations project this gives a simple algorithm for computing the associated variety of the annihilator of an irreducible Harish-Chandra module (submitted).
  

Work in Progress:

• Invariant theory of the enveloping algebra.  A description of primary and secondary $K$-invariants in the enveloping algebra of a symmetric pair.
• Orbit structure of Dynkin-Kostant spaces, with Alfred Noël.  Full orbit decompositions and closure orderings for Dynkin-Kostant spaces arising from Z-graded complex reductive Lie algebras.
• Limited-depth Littlewood-Richardson cones and toric degeneration of the centralizer of a maximal compact in SU(p,q), with Sangjib Kim.  A study of the centralizer of a maximal compact subgroup in the universal enveloping algebra of su(p,q), showing, in particular, that this algebra admits a SAGBI degeneration to the toric algebra associated with a polyhedral cone constructed in a straightforward manner from limited-depth Littlewood-Richardson cones, and utilizing a simplicial decomposition of the depth three LR cone to give explicit generators for U(g)^K in the cases g=su(p,2) and su(p,3).
• Conormality of nilpotent orbits in classical Lie Algebras, with Hervé Sabourin.  Classification of nilpotent orbits whose centralizers admit complementary subalgebras.
• Conormality of nilpotent orbits in exceptional Lie Algebras, with Alfred Noël.  Computer algebra project to find subalgebras complementary to the centralizers of nilpotent elements in exceptional Lie algebras.
• K-spherical flag varieties and multiplicity-free branching rules.  An explicit determination of covariant algebras for certain affine spherical cones, leading to combinatorial (positive-sum) branching rules for the corresponding symmetric pairs.
• A Macaulay2 package for computation of invariants of reductive group actions.  An implementation of Derksen's algorithm for (infinite) reductive algebraic groups.
  

Classes:

Currently (spring 2012) I am teaching:

• Math 260 (Linear Algebra I).

Contact Information:

Steven Glenn Jackson
Department of Mathematics
University of Massachusetts
100 Morrissey Boulevard
Boston, MA 02125-3393

Tel.: (617) 287-6469
Fax: (617) 287-6433
E-mail: jackson@math.umb.edu
Office: Sci-3-082
Drop-in office hours (fall 2011):  Tuesdays and Thursdays, 2:00 p.m.--3:00 p.m.
Appointments: If my drop-in hours are not convenient for you, you may schedule an appointment with me, at least 24 hours in advance, by using my Gmail calendar. (If necessary, first go here to create a Gmail account.) If none of the available times works for you, please try to ask your question by e-mail, or contact the Math office (617-287-6440) for information on other instructors' office hours.

Links:

• My CV.
• The Mathematics Club at UMass Boston.
• The Mathematical Gazette, a weekly guide to mathematics happenings in the Boston area.
• The UMB Mathematics Department seminar schedule.
• MathSciNet, the American Mathematical Society's search engine for Mathematical Reviews.
• The ArXiv, my favorite preprint server.
• Healey Library at UMB.