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Events for February – June 2008

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Feb. 05, 2008 2pm (MATH 350)	Lie Theory Richard Green (CU) X The universal enveloping algebra of a Lie algebra is an associative algebra with the same representation theory. This sequence of talks will be a (possibly intermittent) mini-course in Lie algebras and enveloping algebras, using only MATH 6130/6140 as a prerequisite. It is recommended that students take notes. Topics to be covered include Lie algebras, enveloping algebras, Hopf algebras, the Serre presentation, the triangular decomposition, Verma modules, simple modules and quantized enveloping algebras. The talks will focus on Lie theory and representation theory, not universal algebra as the title might imply. Introduction to universal enveloping algebras III
Feb. 12, 2008 2pm (MATH 350)	Lie Theory Richard Green (CU) X The universal enveloping algebra of a Lie algebra is an associative algebra with the same representation theory. This sequence of talks will be a (possibly intermittent) mini-course in Lie algebras and enveloping algebras, using only MATH 6130/6140 as a prerequisite. It is recommended that students take notes. Topics to be covered include Lie algebras, enveloping algebras, Hopf algebras, the Serre presentation, the triangular decomposition, Verma modules, simple modules and quantized enveloping algebras. The talks will focus on Lie theory and representation theory, not universal algebra as the title might imply. Introduction to universal enveloping algebras IV
Feb. 19, 2008 2pm (MATH 350)	Lie Theory Richard Green (CU) X We will introduce the Serre presentation of the enveloping algebra associated to a Lie algebra, and use it to construct the r-fold tensor power of the natural representation of the Lie algebra of gln. This provides the framework in which to discuss Schur-Weyl duality and Schur algebras. We will present a result of Doty and Giaquinto explaining how to extend the Serre presentation to a presentation of the Schur algebra S(n, r). Schur-Weyl Duality
Mar. 04, 2008 2pm (MATH 350)	Lie Theory Richard Green (CU) X We introduce the triangular decomposition of the universal enveloping algebra associated to a simple Lie algebra over the complex numbers. Using this, we can introduce the notions of highest weight vectors and highest weight modules. We will describe the construction of some specific highest weight modules, including Verma modules and simple modules. Highest weight modules for universal enveloping algebras I
Mar. 11, 2008 2pm (MATH 350)	Lie Theory Richard Green (CU) X We introduce the triangular decomposition of the universal enveloping algebra associated to a simple Lie algebra over the complex numbers. Using this, we can introduce the notions of highest weight vectors and highest weight modules. We will describe the construction of some specific highest weight modules, including Verma modules and simple modules. Highest weight modules for universal enveloping algebras II
Mar. 18, 2008 2pm (MATH 350)	Lie Theory Richard Green (CU) X We introduce the triangular decomposition of the universal enveloping algebra associated to a simple Lie algebra over the complex numbers. Using this, we can introduce the notions of highest weight vectors and highest weight modules. We will describe the construction of some specific highest weight modules, including Verma modules and simple modules. Highest weight modules for universal enveloping algebras III
Apr. 15, 2008 2pm (MATH 350)	Lie Theory Hugh Denoncourt (CU) X Commutation classes are equivalence classes on words that are equivalent by a sequence of commutation relations; that is to say, relations of the form xy=yx. They provide a framework for studying the combinatorial properties of reduced expressions for Coxeter group elements (or for studying the associated roots). We show that the commutation classes for a reduced word of a permutation are in correspondence with certain tableaux and demonstrate interesting features of this correspondence. A tableaux correspondence for commutation classes I
Apr. 22, 2008 2pm (MATH 350)	Lie Theory Hugh Denoncourt (CU) X Commutation classes are equivalence classes on words that are equivalent by a sequence of commutation relations; that is to say, relations of the form xy=yx. They provide a framework for studying the combinatorial properties of reduced expressions for Coxeter group elements (or for studying the associated roots). We show that the commutation classes for a reduced word of a permutation are in correspondence with certain tableaux and demonstrate interesting features of this correspondence. A tableaux correspondence for commutation classes II
Apr. 29, 2008 2pm (MATH 350)	Lie Theory Dan Daly (DU) X Recent work by Tenner has given interest in studying reduced decompositions of permutations with few repetitions. Tenner has shown if a permutation has a reduced decomposition with no repeated elements then one can say quite a bit about the downset of that permutation in the Bruhat Order and about which patterns the permutation avoids. We try to extend these results a bit by looking at permutations with one and two repeated elements. Using such reduced decompositions we obtain new counts for certain classes of pattern-avoiding permutations. Reduced decompositions with few repetitions and permutation patterns

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