"Ring of symmetric functions"의 두 판 사이의 차이
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| 44번째 줄: | 44번째 줄: | ||
(10) Polynomial functors of vector spaces | (10) Polynomial functors of vector spaces | ||
| − | (11)underlying space of algebra of Bosons in 1-dim | + | (11) underlying space of algebra of Bosons in 1-dim |
2012년 8월 25일 (토) 13:16 판
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