The goal of this talk is to

learn how to use symmetric functions
learn how to use affine Stanley symmetric functions
use the set-up that Jason introduced to make them live in the right space

Getting Started: Symmetric functions

{{{id=18| SF = SymmetricFunctions(QQ); SF.inject_shorthands(); /// }}} {{{id=19| h /// }}} {{{id=20| e /// }}} {{{id=21| s /// }}} {{{id=22| m /// }}} {{{id=23| m(s[2,1]) /// }}} {{{id=24| s(m[3,1]) /// }}} {{{id=25| ks3 = kSchurFunctions(QQ, 3); ks3 /// }}} {{{id=27| ks3.base_ring() /// }}} {{{id=28| s = SFASchur(ks3.base_ring()) /// }}} {{{id=26| s(ks3[3,3,1]) /// }}} {{{id=29| ks3 = kSchurFunctions(QQ, 3, 1); ks3 /// }}} {{{id=30| s = SFASchur(ks3.base_ring()) /// }}} {{{id=31| s(ks3[3,3,1]) /// }}} {{{id=32| ks3(ks3[3,1]*ks3[2,1]) /// }}}

Affine Stanley symmetric functions

implemented by Steve Pon, Nicolas Thiery, ...

{{{id=34| W = WeylGroup(['A', 3, 1]) w = W.from_reduced_word([3,1,2,0,3,1,0]) w /// }}} {{{id=35| w.stanley_symmetric_function() /// }}} {{{id=36| w.stanley_symmetric_function?? /// }}} {{{id=37| w.stanley_symmetric_function_as_polynomial?? /// }}} {{{id=38| W = WeylGroup(['C',3,1]) W.from_reduced_word([0,2,1,0]).stanley_symmetric_function() /// }}} {{{id=33| W = WeylGroup(['B',3,1]) W.from_reduced_word([3,2,1]).stanley_symmetric_function() /// }}} {{{id=41| mu = Partition([2,1]) mu.from_kbounded_to_grassmannian(2) /// }}} {{{id=39| def dual_k_schur(mu, k): return Partition(mu).from_kbounded_to_grassmannian(k).stanley_symmetric_function() /// }}} {{{id=42| dual_k_schur([3,1],3) /// }}} {{{id=44| p = SFAPower(QQ) p(dual_k_schur([3,1],3)) /// }}}

Quotient Space

Now we let the Stanley symmetric functions live in the quotient space

{{{id=12| class SFQuotient_basis(CombinatorialFreeModule): def __init__(self, R, k, basis_name): self._basis = basis_name self._prefix = basis_name CombinatorialFreeModule.__init__(self, R, Partitions(NonNegativeIntegers(), max_part=k), category = GradedHopfAlgebrasWithBasis(R)) self.k = k self.SF = SymmetricFunctions(R) /// }}} {{{id=7| class SFQuotient_basis_classical(SFQuotient_basis): def __init__(self, R, k, classical): assert classical in ['m','p'] prefix = classical + classical SFQuotient_basis.__init__(self, R, k, prefix) if classical=='m': self._ambient = self.SF.monomial() else: self._ambient = self.SF.power() self._module_morphism(self._ambient.monomial).register_as_coercion() self._ambient._module_morphism(self.retract_on_basis, codomain=self).register_as_coercion() def retract_on_basis(self, mu): return self.monomial(mu) if (len(mu) == 0 or mu[0] <= self.k) else self.zero() /// }}} {{{id=6| class Dual_ks(SFQuotient_basis): def __init__(self, R, k): self._ambient = SFQuotient_basis_classical(R, k, 'm') SFQuotient_basis.__init__(self, R, k,'dk') lift = self._module_morphism(self.lift_on_basis, codomain=self._ambient, triangular='upper') lift.register_as_coercion() (~lift).register_as_coercion() def lift_on_basis(self, mu): return self._ambient(mu.from_kbounded_to_grassmannian(self.k). stanley_symmetric_function()) /// }}} {{{id=13| SF = SymmetricFunctions(QQ); SF.inject_shorthands() mm = SFQuotient_basis_classical(QQ, 3, 'm') # m basis of quotient pp = SFQuotient_basis_classical(QQ, 3, 'p') # p basis of quotient dk3 = Dual_ks(QQ, 3) # dual k-Schur basis of quotient /// }}} {{{id=3| P53 = Partitions(5, max_part=3) for mu in P53: # Expand dual k-Schur functions into power sums print p(dk3(mu)) /// }}} {{{id=48| for mu in P53: # Expand dual k-Schurs into power sums of the quotient print pp(dk3(mu)) /// }}} {{{id=4| for mu in P53: print dk3(p(mu)) # Expand power sums into dual k-Schurs /// }}} {{{id=49| for mu in P53: print dk3(pp(mu)) # Expand power sums of quotient into dual k-Schurs /// }}} {{{id=10| for mu in P53: print m(dk3(mu)) # Expand dual k-Schurs into monomials /// }}} {{{id=45| for mu in P53: print dk3(m(mu)) # Expand monomials into dual k-Schurs /// }}} {{{id=11| for mu in P53: print pp(p(m(mm(dk3(mu))))) /// }}} {{{id=14| for mu in Partitions(5): print dk3(s(mu)) /// }}}