The single-oscillator Heisenberg algebra arises in quantum mechanics, and is seemingly less sophisticated than the algebras (infinite-oscillator, Virasoro) arising in quantum field theory. It is isomorphic to the algebra generated over C by non-commuting operators x and D (the derivative with respect to x), acting on any suitable space of functions. `Ordering identities' in this algebra include the rewriting of a word (a specified product of x's and D's) as a combination of normally ordered words: ones in which D's appear to the right of x's. In the relatively easy `derivation' case, when each word is based on (a power of) a word containing only a single D, the coefficients turn out to be Stirling numbers, which have a combinatorial interpretation. Ordering identities can be much more complicated, though any such identity can be viewed as an element of a two-sided ideal in the free associative algebra generated by x and D. Exploring the space of ordering identities, we introduce triangles of recursively defined generalized Stirling and Eulerian numbers, and identities in which these numbers appear as coefficients. Many of these triangles have no easy combinatorial interpretation, but satisfy triangular recurrences from which generating functions can be computed by the method of characteristics.
Combinatorics in the single-oscillator Heisenberg algebra
Tue, Apr. 22 2:30pm (MATH 3…
Lie Theory
Juan Villarreal (University of Colorado Boulder)
X
In this talk, we will show how to associate certain varieties to algebras and modules, called associated varieties. Then, we will explain how some geometric properties of these varieties have deep implications in representation theory. Finally, we will explain how some of these relations also holds in infinite dimensions.
Filtrations and associated varieties.
Tue, Apr. 22 3:30pm (MATH 3…
Topology
Alexander Waugh (University of Washington)
X
In this talk, I willI introduce a general framework for how we can understand properties of power operations via "Eulerian sequences". I will also discuss how various properties of these operations are encoded by these sequences (e.g. geometric fixed points, Cartan formula, etc...). When the group of equivariance is trivial or has order two, all known Steenrod and Dyer-Lashof operations are recovered in this framework. As a final application, I will show how to construct new nonzero mod p Steenrod and Dyer-Lashof operations for every finite group. This is based on joint work with Prasit Bhattacharya and Foling Zou.
Combinatorics in the single-oscillator Heisenberg algebra