Ring of symmetric functions

structure on ring of symmetric functions S

 

1. commutative algebra
2. cocommutative coalgebra
3. antipode involutions
4. symmetric bilinear form <,> algebra structure dual to coalgebra structure
5. partial order \geq
6. 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