Ring of symmetric functions
structure on ring of symmetric functions S
- commutative algebra
- cocommutative coalgebra
- antipode involutions
- symmetric bilinear form <,> algebra structure dual to coalgebra structure
- partial order \geq
- lots of bases
1,2,3 => commutative, cocommutative Hopf algebra, coordinate ring of a commutative group scheme
S\otimes \mathbb{Q} is UEA of a Lie algebra
list of places where algebra S of symmetric functions turns up
(1) ring of symmetric functions
(2) representation theory of symmetric group S_n
(3) representation theory of general linear group Gl_n
(4) homology of BU (classifying space for vector bundles)
(5) Cohomology of Grassmannians
(6) Schubert calculus
(7) universal \lambda ring on 1-generator
(8) coordinate ring of group scheme of power series 1+e_1x+e_2x^2+\cdots
(9) Hall algebra of finite abelian p-groups
(10) Polynomial functors of vector spaces
(11) underlying space of algebra of Bosons in 1-dim