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r""" This file contains doctests for the Chapter "k-Schur function primer" for the book "k-Schur functions and affine Schubert calculus" by Thomas Lam, Luc Lapointe, Jennifer Morse, Anne Schilling, Mark Shimozono, and Mike Zabrocki, :arxiv:`1301.3569`. The code was written by Anne Schilling and Mike Zabrocki, 2012 and 2013.
IF IT BECOMES NECESSARY TO CHANGE ANY TESTS IN THIS FILE, THERE NEEDS TO BE A ONE-YEAR DEPRECATION PERIOD. ALSO, PLEASE IN THIS CASE CONTACT Anne Schilling (anne@math.ucdavis.edu) AND Mike Zabrocki (zabrocki@mathstat.yorku.ca) REGARDING THE CHANGES! """
""" Sage example in ./kschurnotes/notes-mike-anne.tex, line 198::
sage: P = Partitions(4); P Partitions of the integer 4 sage: P.list() [[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 205::
sage: [p for p in P] [[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 210::
sage: la=Partition([2,2]); mu=Partition([3,1]) sage: mu.dominates(la) True
Sage example in ./kschurnotes/notes-mike-anne.tex, line 216::
sage: ord = lambda x,y: y.dominates(x) sage: P = Poset([Partitions(6), ord], facade=True) sage: H = P.hasse_diagram()
Sage example in ./kschurnotes/notes-mike-anne.tex, line 228::
sage: la=Partition([4,3,3,3,2,2,1]) sage: la.conjugate() [7, 6, 4, 1] sage: la.k_split(4) [[4], [3, 3], [3, 2], [2, 1]]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 236::
sage: p = SkewPartition([[2,1],[1]]) sage: p.is_connected() False
Sage example in ./kschurnotes/notes-mike-anne.tex, line 334::
sage: la = Partition([4,3,3,3,2,2,1]) sage: kappa = la.k_skew(4); kappa [12, 8, 5, 5, 2, 2, 1] / [8, 5, 2, 2]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 340::
sage: kappa.row_lengths() [4, 3, 3, 3, 2, 2, 1]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 345::
sage: tau = Core([12,8,5,5,2,2,1],5) sage: mu = tau.to_bounded_partition(); mu [4, 3, 3, 3, 2, 2, 1] sage: mu.to_core(4) [12, 8, 5, 5, 2, 2, 1]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 353::
sage: Cores(3,6).list() [[6, 4, 2], [5, 3, 1, 1], [4, 2, 2, 1, 1], [3, 3, 2, 2, 1, 1]]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 398::
sage: W = WeylGroup(['A',4,1]) # long time (5.47 s, 2013) sage: S = W.simple_reflections() # long time sage: [s.reduced_word() for s in S] # long time [[0], [1], [2], [3], [4]]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 406::
sage: w = W.an_element(); w # long time [ 2 0 0 1 -2] [ 2 0 0 0 -1] [ 1 1 0 0 -1] [ 1 0 1 0 -1] [ 1 0 0 1 -1] sage: w.reduced_word() # long time [0, 1, 2, 3, 4] sage: w = W.from_reduced_word([2,1,0]) # long time sage: w.is_affine_grassmannian() # long time True
Sage example in ./kschurnotes/notes-mike-anne.tex, line 464::
sage: c = Core([7,3,1],5) sage: c.affine_symmetric_group_simple_action(2) [8, 4, 1, 1] sage: c.affine_symmetric_group_simple_action(0) [7, 3, 1]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 474::
sage: k=4; length=3 sage: W = WeylGroup(['A',k,1]) sage: G = W.affine_grassmannian_elements_of_given_length(length) sage: [w.reduced_word() for w in G] [[2, 1, 0], [4, 1, 0], [3, 4, 0]]
sage: C = Cores(k+1,length) sage: [c.to_grassmannian().reduced_word() for c in C] [[2, 1, 0], [4, 1, 0], [3, 4, 0]]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 543::
sage: la = Partition([4,3,3,3,2,2,1]) sage: c = la.to_core(4); c [12, 8, 5, 5, 2, 2, 1] sage: W = WeylGroup(['A',4,1]) sage: w = W.from_reduced_word([4,1,0,2,1,4,3,2,0,4,3,1,0,4,3,2,1,0]) sage: c.to_grassmannian() == w True
Sage example in ./kschurnotes/notes-mike-anne.tex, line 643::
sage: A = AffinePermutationGroup(['A',2,1]) sage: w = A([-2,0,8]) sage: w.reduced_word() [1, 0, 2, 1, 0] sage: w.to_core() [5, 3, 1]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 767::
sage: la = Partition([4,3,3,3,2,2,1]) sage: la.k_conjugate(4) [3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 1065::
sage: c = Core([3,1,1],3) sage: c.weak_covers() [[4, 2, 1, 1]] sage: c.strong_covers() [[5, 3, 1], [4, 2, 1, 1], [3, 2, 2, 1, 1]]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 1073::
sage: kappa = Core([4,1],4) sage: tau = Core([2,1],4) sage: tau.weak_le(kappa) False sage: tau.strong_le(kappa) True
Sage example in ./kschurnotes/notes-mike-anne.tex, line 1082::
sage: C = sum(([c for c in Cores(4,m)] for m in range(7)),[]) sage: ord = lambda x,y: x.weak_le(y) sage: P = Poset([C, ord], cover_relations = False) # long time (3.99 s, 2013) sage: H = P.hasse_diagram() # long time sage: view(H) # not tested
Sage example in ./kschurnotes/notes-mike-anne.tex, line 1253::
sage: Sym = SymmetricFunctions(QQ) sage: h = Sym.homogeneous() sage: m = Sym.monomial()
Sage example in ./kschurnotes/notes-mike-anne.tex, line 1259::
sage: f = h[3,1]+h[2,2] sage: m(f) 10*m[1, 1, 1, 1] + 7*m[2, 1, 1] + 5*m[2, 2] + 4*m[3, 1] + 2*m[4]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 1266::
sage: f.scalar(h[2,1,1]) 7 sage: m(f).coefficient([2,1,1]) 7
Sage example in ./kschurnotes/notes-mike-anne.tex, line 1274::
sage: p = Sym.power() sage: e = Sym.elementary() sage: sum( (-1)**(i-1)*e[4-i]*p[i] for i in range(1,4) ) - p[4] 4*e[4] sage: sum( (-1)**(i-1)*p[i]*e[4-i] for i in range(1,4) ) - p[4] 1/6*p[1, 1, 1, 1] - p[2, 1, 1] + 1/2*p[2, 2] + 4/3*p[3, 1] - p[4]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 1327::
sage: Sym = SymmetricFunctions(QQ) sage: s = Sym.schur() sage: m = Sym.monomial() sage: h = Sym.homogeneous() sage: m(s[1,1,1]) m[1, 1, 1] sage: h(s[1,1,1]) h[1, 1, 1] - 2*h[2, 1] + h[3]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 1338::
sage: p = Sym.power() sage: s = Sym.schur() sage: p(s[1,1,1]) 1/6*p[1, 1, 1] - 1/2*p[2, 1] + 1/3*p[3] sage: p(s[2,1]) 1/3*p[1, 1, 1] - 1/3*p[3] sage: p(s[3]) 1/6*p[1, 1, 1] + 1/2*p[2, 1] + 1/3*p[3]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 1349::
sage: s[2,1].scalar(s[1,1,1]) 0 sage: s[2,1].scalar(s[2,1]) 1
Sage example in ./kschurnotes/notes-mike-anne.tex, line 1517::
sage: Sym = SymmetricFunctions(FractionField(QQ['t'])) sage: Qp = Sym.hall_littlewood().Qp() sage: Qp.base_ring() Fraction Field of Univariate Polynomial Ring in t over Rational Field sage: s = Sym.schur() sage: s(Qp[1,1,1]) s[1, 1, 1] + (t^2+t)*s[2, 1] + t^3*s[3]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 1530::
sage: t = Qp.t sage: s[2,1].scalar(s[3].theta_qt(t,0)) t^2 - t
Sage example in ./kschurnotes/notes-mike-anne.tex, line 1536::
sage: s(Qp([1,1])).hl_creation_operator([3]) s[3, 1, 1] + t*s[3, 2] + (t^2+t)*s[4, 1] + t^3*s[5] sage: s(Qp([3,1,1])) s[3, 1, 1] + t*s[3, 2] + (t^2+t)*s[4, 1] + t^3*s[5]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 1568::
sage: Sym = SymmetricFunctions(FractionField(QQ['q,t'])) sage: Mac = Sym.macdonald() sage: H = Mac.H() sage: s = Sym.schur() sage: for la in Partitions(3): ....: print("H {} = {}".format(la, s(H(la)))) H [3] = q^3*s[1, 1, 1] + (q^2+q)*s[2, 1] + s[3] H [2, 1] = q*s[1, 1, 1] + (q*t+1)*s[2, 1] + t*s[3] H [1, 1, 1] = s[1, 1, 1] + (t^2+t)*s[2, 1] + t^3*s[3]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 1581::
sage: Sym = SymmetricFunctions(FractionField(QQ['t'])) sage: Mac = Sym.macdonald(q=0) sage: H = Mac.H() sage: s = Sym.schur() sage: for la in Partitions(3): ....: print("H {} = {}".format(la, s(H(la)))) H [3] = s[3] H [2, 1] = s[2, 1] + t*s[3] H [1, 1, 1] = s[1, 1, 1] + (t^2+t)*s[2, 1] + t^3*s[3] sage: Qp = Sym.hall_littlewood().Qp() sage: s(Qp[1, 1, 1]) s[1, 1, 1] + (t^2+t)*s[2, 1] + t^3*s[3]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 1596::
sage: Sym = SymmetricFunctions(FractionField(QQ['q'])) sage: Mac = Sym.macdonald(t=0) sage: H = Mac.H() sage: s = Sym.schur() sage: for la in Partitions(3): ....: print("H {} = {}".format(la, s(H(la)))) H [3] = q^3*s[1, 1, 1] + (q^2+q)*s[2, 1] + s[3] H [2, 1] = q*s[1, 1, 1] + s[2, 1] H [1, 1, 1] = s[1, 1, 1]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 1717::
sage: t = Tableau([[1,1,1,2,3,7],[2,2,3,5],[3,4],[4,5],[6]]) sage: t.charge() 9
Sage example in ./kschurnotes/notes-mike-anne.tex, line 1881::
sage: la = Partition([2,2]) sage: la.k_conjugate(2).conjugate() [4] sage: la = Partition([2,1,1]) sage: la.k_conjugate(2).conjugate() [3, 1] sage: la = Partition([1,1,1,1]) sage: la.k_conjugate(2).conjugate() [2, 2]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 1893::
sage: Sym = SymmetricFunctions(FractionField(QQ['t'])) sage: ks = Sym.kschur(2) sage: ks[2,2].omega_t_inverse() 1/t^2*ks2[1, 1, 1, 1] sage: ks[2,1,1].omega_t_inverse() 1/t*ks2[2, 1, 1] sage: ks[1,1,1,1].omega_t_inverse() 1/t^2*ks2[2, 2]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 1904::
sage: Sym = SymmetricFunctions(FractionField(QQ['q,t'])) sage: H = Sym.macdonald().H() sage: ks = Sym.kschur(2) sage: ks(H[2,2]) q^2*ks2[1, 1, 1, 1] + (q*t+q)*ks2[2, 1, 1] + ks2[2, 2] sage: ks(H[2,1,1]) q*ks2[1, 1, 1, 1] + (q*t^2+1)*ks2[2, 1, 1] + t*ks2[2, 2] sage: ks(H[1,1,1,1]) ks2[1, 1, 1, 1] + (t^3+t^2)*ks2[2, 1, 1] + t^4*ks2[2, 2]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 2174::
sage: SemistandardTableaux([5,2],[4,2,1]).list() [[[1, 1, 1, 1, 2], [2, 3]], [[1, 1, 1, 1, 3], [2, 2]]]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 2179::
sage: P = Partitions(4) sage: P.list() [[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]] sage: n = P.cardinality(); n 5 sage: K = matrix(QQ,n,n, ....: [[SemistandardTableaux(la,mu).cardinality() ....: for mu in P] for la in P]) sage: K [1 1 1 1 1] [0 1 1 2 3] [0 0 1 1 2] [0 0 0 1 3] [0 0 0 0 1]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 2376::
sage: T = WeakTableaux(6, [5,3], [4,3,1]) sage: T.list() [[[1, 1, 1, 1, 3], [2, 2, 2]], [[1, 1, 1, 1, 2], [2, 2, 3]]]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 2382::
sage: k = 3 sage: c = Core([5,2,1], k+1) sage: la = c.to_bounded_partition(); la [3, 2, 1] sage: for mu in Partitions(la.size(), max_part = 3): ....: T = WeakTableaux(k, c, mu) ....: print("weight {}".format(mu)) ....: print(T.list()) ....: weight [3, 3] [] weight [3, 2, 1] [[[1, 1, 1, 2, 2], [2, 2], [3]]] weight [3, 1, 1, 1] [[[1, 1, 1, 2, 4], [2, 4], [3]], [[1, 1, 1, 2, 3], [2, 3], [4]]] weight [2, 2, 2] [[[1, 1, 2, 2, 3], [2, 3], [3]]] weight [2, 2, 1, 1] [[[1, 1, 2, 2, 4], [2, 4], [3]], [[1, 1, 2, 2, 3], [2, 3], [4]]] weight [2, 1, 1, 1, 1] [[[1, 1, 3, 4, 5], [2, 5], [3]], [[1, 1, 2, 3, 5], [3, 5], [4]], [[1, 1, 2, 3, 4], [3, 4], [5]]] weight [1, 1, 1, 1, 1, 1] [[[1, 3, 4, 5, 6], [2, 6], [4]], [[1, 2, 4, 5, 6], [3, 6], [4]], [[1, 2, 3, 4, 6], [4, 6], [5]], [[1, 2, 3, 4, 5], [4, 5], [6]]]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 2487::
sage: Sym = SymmetricFunctions(QQ) sage: ks = Sym.kschur(3,t=1) sage: h = Sym.homogeneous() sage: for mu in Partitions(7, max_part =3): ....: print(h(ks(mu))) h[3, 3, 1] h[3, 2, 2] - h[3, 3, 1] h[3, 2, 1, 1] - h[3, 2, 2] h[3, 1, 1, 1, 1] - 2*h[3, 2, 1, 1] + h[3, 3, 1] h[2, 2, 2, 1] - h[3, 2, 1, 1] - h[3, 2, 2] + h[3, 3, 1] h[2, 2, 1, 1, 1] - 2*h[2, 2, 2, 1] - h[3, 1, 1, 1, 1] + 2*h[3, 2, 1, 1] + h[3, 2, 2] - h[3, 3, 1] h[2, 1, 1, 1, 1, 1] - 3*h[2, 2, 1, 1, 1] + 2*h[2, 2, 2, 1] + h[3, 2, 1, 1] - h[3, 2, 2] h[1, 1, 1, 1, 1, 1, 1] - 4*h[2, 1, 1, 1, 1, 1] + 4*h[2, 2, 1, 1, 1] + 2*h[3, 1, 1, 1, 1] - 4*h[3, 2, 1, 1] + h[3, 3, 1]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 2608::
sage: ks6 = Sym.kschur(6,t=1) sage: ks6(h[4,3,1]) ks6[4, 3, 1] + ks6[4, 4] + ks6[5, 2, 1] + 2*ks6[5, 3] + ks6[6, 1, 1] + ks6[6, 2]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 2617::
sage: Sym = SymmetricFunctions(QQ) sage: ks = Sym.kschur(3,t=1) sage: ks.realization_of() 3-bounded Symmetric Functions over Rational Field with t=1 sage: s = Sym.schur() sage: s.realization_of() Symmetric Functions over Rational Field
Sage example in ./kschurnotes/notes-mike-anne.tex, line 2658::
sage: k = 6 sage: weight = Partition([4,3,1]) sage: for la in Partitions(weight.size(), max_part = k): ....: if la.dominates(weight): ....: print(la) ....: T = WeakTableaux(k, la, weight, representation = 'bounded') ....: print(T.list()) [6, 2] [[[1, 1, 1, 1, 2, 2], [2, 3]]] [6, 1, 1] [[[1, 1, 1, 1, 2, 2], [2], [3]]] [5, 3] [[[1, 1, 1, 1, 3], [2, 2, 2]], [[1, 1, 1, 1, 2], [2, 2, 3]]] [5, 2, 1] [[[1, 1, 1, 1, 2], [2, 2], [3]]] [4, 4] [[[1, 1, 1, 1], [2, 2, 2, 3]]] [4, 3, 1] [[[1, 1, 1, 1], [2, 2, 2], [3]]]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 2740::
sage: mu = Partition([3,2,1]) sage: c = mu.to_core(3) sage: w = c.to_grassmannian() sage: w.stanley_symmetric_function() 4*m[1, 1, 1, 1, 1, 1] + 3*m[2, 1, 1, 1, 1] + 2*m[2, 2, 1, 1] + m[2, 2, 2] + 2*m[3, 1, 1, 1] + m[3, 2, 1] sage: w.reduced_words() [[2, 0, 3, 2, 1, 0], [0, 2, 3, 2, 1, 0], [0, 3, 2, 3, 1, 0], [0, 3, 2, 1, 3, 0]]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 2752::
sage: Sym = SymmetricFunctions(QQ) sage: Q3 = Sym.kBoundedQuotient(3,t=1) sage: F3 = Q3.affineSchur() sage: m = Q3.kmonomial() sage: m(F3([3,2,1])) 4*m3[1, 1, 1, 1, 1, 1] + 3*m3[2, 1, 1, 1, 1] + 2*m3[2, 2, 1, 1] + m3[2, 2, 2] + 2*m3[3, 1, 1, 1] + m3[3, 2, 1]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 2799::
sage: Sym = SymmetricFunctions(QQ) sage: Q3 = Sym.kBoundedQuotient(3,t=1) sage: F3 = Q3.affineSchur() sage: h = Sym.homogeneous() sage: f = F3[3,2,1]*h[1]; f F3[3, 1, 1, 1, 1] + 3*F3[3, 2, 1, 1] + F3[3, 2, 2] + 2*F3[3, 3, 1]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 2810::
sage: c = Partition([3,2,1]).to_core(3) sage: for p in sorted(f.support()): # Sorted for consistant doctest ordering ....: print("{} {}".format(p, SkewPartition([p.to_core(3).to_partition(),c.to_partition()]))) ....: [3, 1, 1, 1, 1] [[5, 2, 1, 1, 1], [5, 2, 1]] [3, 2, 1, 1] [[6, 3, 1, 1], [5, 2, 1]] [3, 2, 2] [[5, 2, 2], [5, 2, 1]] [3, 3, 1] [[7, 4, 1], [5, 2, 1]]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 2976::
sage: T = StrongTableau([[-1,-1,-2,-3],[-2,3,-3,4],[2,3],[-3,-4]], 3) sage: T.to_transposition_sequence() [[-2, -1], [3, 4], [0, 2], [-3, -2], [2, 3], [-1, 0], [1, 2], [0, 1]] sage: T.intermediate_shapes() [[], [2], [3, 1, 1], [4, 3, 2, 1], [4, 4, 2, 2]] sage: [T.content_of_marked_head(v+1) for v in range(8)] [0, 1, -1, 2, -3, 1, 3, -2] sage: T.left_action([0,1]) [[-1, -1, -2, -3, 5], [-2, 3, -3, 4], [2, 3, -5], [-3, -4], [5]]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 2999::
sage: ST = StrongTableaux(3, [6,3,1,1], [4,2,1]); ST Set of strong 3-tableaux of shape [6, 3, 1, 1] and of weight (4, 2, 1) sage: ST.list() [[[-1, -1, -1, -1, 2, 2], [1, -2, -2], [-3], [3]], [[-1, -1, -1, -1, 2, -2], [1, -2, 2], [-3], [3]], [[-1, -1, -1, -1, -2, -2], [1, 2, 2], [-3], [3]], [[-1, -1, -1, -1, 2, 3], [1, -2, 3], [-2], [-3]], [[-1, -1, -1, -1, 2, 3], [1, -2, -3], [-2], [3]], [[-1, -1, -1, -1, 2, -3], [1, -2, 3], [-2], [3]], [[-1, -1, -1, -1, -2, 3], [1, 2, 3], [-2], [-3]], [[-1, -1, -1, -1, -2, 3], [1, 2, -3], [-2], [3]], [[-1, -1, -1, -1, -2, -3], [1, 2, 3], [-2], [3]]] sage: ks = SymmetricFunctions(QQ).kschur(3,1) sage: m = SymmetricFunctions(QQ).m() sage: m(ks[3,2,1,1]).coefficient([4,2,1]) 9
Sage example in ./kschurnotes/notes-mike-anne.tex, line 3243::
sage: W = WeylGroup(['A',3,1]) sage: [w.reduced_word() for w in W.pieri_factors()] [[], [0], [1], [2], [3], [1, 0], [2, 0], [0, 3], [2, 1], [3, 1], [3, 2], [2, 1, 0], [1, 0, 3], [0, 3, 2], [3, 2, 1]]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 3251::
sage: A = NilCoxeterAlgebra(WeylGroup(['A',3,1]), prefix = 'A') sage: A.homogeneous_noncommutative_variables([2]) A[1,0] + A[2,0] + A[0,3] + A[3,2] + A[3,1] + A[2,1]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 3258::
sage: A.k_schur_noncommutative_variables([2,2]) A[0,3,1,0] + A[3,1,2,0] + A[1,2,0,1] + A[3,2,0,3] + A[2,0,3,1] + A[2,3,1,2]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 3265::
sage: Sym = SymmetricFunctions(ZZ) sage: ks = Sym.kschur(5,t=1) sage: ks[2,1]*ks[2,1] ks5[2, 2, 1, 1] + ks5[2, 2, 2] + ks5[3, 1, 1, 1] + 2*ks5[3, 2, 1] + ks5[3, 3] + ks5[4, 2]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 3536::
sage: la = Partition([3,2,1,1]) sage: la.k_atom(4) [[[1, 1, 1], [2, 2], [3], [4]], [[1, 1, 1, 4], [2, 2], [3]]]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 3639::
sage: s = SymmetricFunctions(QQ['t']).schur() sage: G1 = s[1] sage: G211 = G1.hl_creation_operator([2,1]); G211 s[2, 1, 1] + t*s[2, 2] + t*s[3, 1] sage: G3211 = G211.hl_creation_operator([3]); G3211 s[3, 2, 1, 1] + t*s[3, 2, 2] + t*s[3, 3, 1] + t*s[4, 1, 1, 1] + (2*t^2+t)*s[4, 2, 1] + t^2*s[4, 3] + (t^3+t^2)*s[5, 1, 1] + 2*t^3*s[5, 2] + t^4*s[6, 1]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 3954::
sage: T = WeakTableau([[1,1,2,3,4,4,5,5,6],[2,3,5,5,6],[3,4,7], ....: [5,6],[6],[7]],4) sage: T.k_charge() 12
Sage example in ./kschurnotes/notes-mike-anne.tex, line 3962::
sage: Sym = SymmetricFunctions(QQ['t'].fraction_field()) sage: Qp = Sym.hall_littlewood().Qp() sage: ks = Sym.kBoundedSubspace(3).kschur() sage: t = ks.base_ring().gen() sage: ks(Qp[3,2,2,1]) ks3[3, 2, 2, 1] + t*ks3[3, 3, 1, 1] + t^2*ks3[3, 3, 2] sage: sum(t^T.k_charge()*ks(la) for la in Partitions(8, max_part=3) ....: for T in WeakTableaux(3,la,[3,2,2,1],representation = 'bounded')) ks3[3, 2, 2, 1] + t*ks3[3, 3, 1, 1] + t^2*ks3[3, 3, 2]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 4055::
sage: t = var('t') sage: for mu in Partitions(5): ....: print("{} {}".format(mu, sum(t^T.spin() for T in StrongTableaux(3,[4,1,1],mu)))) [5] 0 [4, 1] t [3, 2] t [3, 1, 1] 2*t + 1 [2, 2, 1] 2*t + 1 [2, 1, 1, 1] 3*t + 3 [1, 1, 1, 1, 1] 4*t + 6 sage: StrongTableaux( 3, [4,1,1], (1,)*5 ).cardinality() 10 sage: StrongTableaux( 3, [4,1,1], (1,)*5 ).list() [[[-1, -2, -3, 4], [-4], [-5]], [[-1, -2, -3, -4], [4], [-5]], [[-1, -2, -3, -5], [-4], [4]], [[-1, -2, 4, -4], [-3], [-5]], [[-1, -2, 4, -5], [-3], [-4]], [[-1, -2, -4, -5], [-3], [4]], [[-1, -3, 4, -4], [-2], [-5]], [[-1, -3, 4, -5], [-2], [-4]], [[-1, -3, -4, -5], [-2], [4]], [[-1, 4, -4, -5], [-2], [-3]]]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 4385::
sage: Sym = SymmetricFunctions(FractionField(QQ['t'])) sage: ks4 = Sym.kschur(4) sage: ks4([3, 1, 1]).hl_creation_operator([1]) (t-1)*ks4[2, 2, 1, 1] + t^2*ks4[3, 1, 1, 1] + t^3*ks4[3, 2, 1] + (t^3-t^2)*ks4[3, 3] + t^4*ks4[4, 1, 1] sage: ks4([3, 1, 1]).hl_creation_operator([2]) t*ks4[3, 2, 1, 1] + t^2*ks4[3, 3, 1] + t^2*ks4[4, 1, 1, 1] + t^3*ks4[4, 2, 1] sage: ks4([3, 1, 1]).hl_creation_operator([3]) ks4[3, 3, 1, 1] + t*ks4[4, 2, 1, 1] + t^2*ks4[4, 3, 1] sage: ks4([3, 1, 1]).hl_creation_operator([4]) ks4[4, 3, 1, 1]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 4456::
sage: Sym = SymmetricFunctions(FractionField(QQ['t'])) sage: ks3 = Sym.kschur(3) sage: ks3([3,2]).omega() Traceback (most recent call last): ... ValueError: t^2*s[1, 1, 1, 1, 1] + t*s[2, 1, 1, 1] + s[2, 2, 1] is not in the image
sage: s = Sym.schur() sage: s(ks3[3,2]) s[3, 2] + t*s[4, 1] + t^2*s[5] sage: t = s.base_ring().gen() sage: invert = lambda x: s.base_ring()(x.subs(t=1/t)) sage: ks3(s(ks3([3,2])).omega().map_coefficients(invert)) 1/t^2*ks3[1, 1, 1, 1, 1]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 4478::
sage: ks3[3,2].omega_t_inverse() 1/t^2*ks3[1, 1, 1, 1, 1]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 4686::
sage: Sym = SymmetricFunctions(FractionField(QQ['t'])) sage: ks3 = Sym.kschur(3) sage: ks3[3,1].coproduct() ks3[] # ks3[3, 1] + ks3[1] # ks3[2, 1] + (t+1)*ks3[1] # ks3[3] + ks3[1, 1] # ks3[2] + ks3[2] # ks3[1, 1] + (t+1)*ks3[2] # ks3[2] + ks3[2, 1] # ks3[1] + (t+1)*ks3[3] # ks3[1] + ks3[3, 1] # ks3[]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 4720::
sage: Sym = SymmetricFunctions(FractionField(QQ['t'])) sage: ks2 = Sym.kschur(2) sage: ks3 = Sym.kschur(3) sage: ks5 = Sym.kschur(5) sage: ks5(ks3[2])*ks5(ks2[1]) ks5[2, 1] + ks5[3] sage: ks5(ks3[2])*ks5(ks2[2,1]) ks5[2, 2, 1] + ks5[3, 1, 1] + (t+1)*ks5[3, 2] + (t+1)*ks5[4, 1] + t*ks5[5]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 4779::
sage: Sym = SymmetricFunctions(FractionField(QQ['t'])) sage: ks3 = Sym.kschur(3) sage: ks4 = Sym.kschur(4) sage: ks5 = Sym.kschur(5) sage: ks4(ks3[3,2,1,1]) ks4[3, 2, 1, 1] + t*ks4[3, 3, 1] + t*ks4[4, 1, 1, 1] + t^2*ks4[4, 2, 1] sage: ks5(ks3[3,2,1,1]) ks5[3, 2, 1, 1] + t*ks5[3, 3, 1] + t*ks5[4, 1, 1, 1] + t^2*ks5[4, 2, 1] + t^2*ks5[4, 3] + t^3*ks5[5, 1, 1]
sage: ks5(ks4[3,2,1,1]) ks5[3, 2, 1, 1] sage: ks5(ks4[4,3,3,2,1,1]) ks5[4, 3, 3, 2, 1, 1] + t*ks5[4, 4, 3, 1, 1, 1] + t^2*ks5[5, 3, 3, 1, 1, 1] sage: ks5(ks4[4,3,3,2,1,1,1]) ks5[4, 3, 3, 2, 1, 1, 1] + t*ks5[4, 3, 3, 3, 1, 1] + t*ks5[4, 4, 3, 1, 1, 1, 1] + t^2*ks5[4, 4, 3, 2, 1, 1] + t^2*ks5[5, 3, 3, 1, 1, 1, 1] + t^3*ks5[5, 3, 3, 2, 1, 1] + t^4*ks5[5, 4, 3, 1, 1, 1]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 4858::
sage: Sym = SymmetricFunctions(FractionField(QQ['q,t'])) sage: H = Sym.macdonald().H() sage: ks = Sym.kschur(3) sage: ks(H[3]) q^3*ks3[1, 1, 1] + (q^2+q)*ks3[2, 1] + ks3[3] sage: ks(H[3,2]) # long time (2.11 s, 2013) q^4*ks3[1, 1, 1, 1, 1] + (q^3*t+q^3+q^2)*ks3[2, 1, 1, 1] + (q^3*t+q^2*t+q^2+q)*ks3[2, 2, 1] + (q^2*t+q*t+q)*ks3[3, 1, 1] + ks3[3, 2] sage: ks(H[3,1,1]) q^3*ks3[1, 1, 1, 1, 1] + (q^3*t^2+q^2+q)*ks3[2, 1, 1, 1] + (q^2*t^2+q^2*t+q*t+q)*ks3[2, 2, 1] + (q^2*t^2+q*t^2+1)*ks3[3, 1, 1] + t*ks3[3, 2]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 4965::
sage: Sym = SymmetricFunctions(QQ) sage: Q3 = Sym.kBoundedQuotient(3,t=1) sage: F = Q3.affineSchur() sage: p = Sym.power() sage: F[2,1]*p[2] -F3[1, 1, 1, 1, 1] - F3[2, 1, 1, 1] + F3[3, 1, 1] + F3[3, 2]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 5028::
sage: R = QQ[I]; z4 = R.zeta(4) sage: Sym = SymmetricFunctions(R) sage: ks3z = Sym.kschur(3,t=z4) sage: ks3 = Sym.kschur(3,t=1) sage: p = Sym.p() sage: p(ks3z[2, 2, 2, 2, 2, 2, 2, 2]) # long time (17s on sage.math, 2013) 1/12*p[4, 4, 4, 4] + 1/4*p[8, 8] - 1/3*p[12, 4] sage: p(ks3[2,2]) 1/12*p[1, 1, 1, 1] + 1/4*p[2, 2] - 1/3*p[3, 1] sage: p(ks3[2,2]).plethysm(p[4]) 1/12*p[4, 4, 4, 4] + 1/4*p[8, 8] - 1/3*p[12, 4]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 5042::
sage: ks3z[3, 3, 3, 3]*ks3z[2, 1] # long time (10s on sage.math, 2013) ks3[3, 3, 3, 3, 2, 1]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 5197::
sage: Sym = SymmetricFunctions(QQ['t'].fraction_field()) sage: Q3 = Sym.kBoundedQuotient(3) sage: dks = Q3.dual_k_Schur() sage: dks[2, 1, 1]*dks[3, 2, 1] # long time (25.7s, 2013) (t^7+t^6)*dks3[2, 1, 1, 1, 1, 1, 1, 1, 1] + (t^4+t^3+t^2)*dks3[2, 2, 2, 1, 1, 1, 1] + (t^3+t^2)*dks3[2, 2, 2, 2, 1, 1] + (t^5+2*t^4+2*t^3+t^2)*dks3[2, 2, 2, 2, 2] + (t^5+2*t^4+t^3)*dks3[3, 1, 1, 1, 1, 1, 1, 1] + (2*t^5+3*t^4+4*t^3+3*t^2+t)*dks3[3, 2, 1, 1, 1, 1, 1] + (2*t^2+t+1)*dks3[3, 2, 2, 1, 1, 1] + (t^4+3*t^3+4*t^2+3*t+1)*dks3[3, 2, 2, 2, 1] + (t^5+t^4+4*t^3+4*t^2+3*t+1)*dks3[3, 3, 1, 1, 1, 1] + (2*t^5+3*t^4+5*t^3+6*t^2+4*t+2)*dks3[3, 3, 2, 1, 1] + (t^4+t^3+3*t^2+2*t+1)*dks3[3, 3, 2, 2] + (t^5+3*t^4+3*t^3+4*t^2+2*t+1)*dks3[3, 3, 3, 1]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 5294::
sage: ks2 = SymmetricFunctions(QQ['t']).kschur(2) sage: HLQp = SymmetricFunctions(QQ['t']).hall_littlewood().Qp() sage: ks2( (HLQp(ks2[1,1])*HLQp(ks2[1])).restrict_parts(2) ) ks2[1, 1, 1] + (-t+1)*ks2[2, 1]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 5303::
sage: dks = SymmetricFunctions(QQ['t']).kBoundedQuotient(2).dks() sage: dks[2,1].coproduct() dks2[] # dks2[2, 1] + (-t+1)*dks2[1] # dks2[1, 1] + dks2[1] # dks2[2] + (-t+1)*dks2[1, 1] # dks2[1] + dks2[2] # dks2[1] + dks2[2, 1] # dks2[]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 5560::
sage: Sym = SymmetricFunctions(QQ) sage: Sym3 = Sym.kBoundedSubspace(3,t=1) sage: Kks3 = Sym3.K_kschur() sage: s = Sym.s() sage: m = Sym.m() sage: s(Kks3[3,1]) s[3] + s[3, 1] + s[4] sage: m(Kks3[3,1]) m[1, 1, 1] + 4*m[1, 1, 1, 1] + m[2, 1] + 3*m[2, 1, 1] + 2*m[2, 2] + m[3] + 2*m[3, 1] + m[4] sage: ks3 = Sym3.kschur() sage: ks3(Kks3[3,1]) ks3[3] + ks3[3, 1] sage: Kks3[3,1]*Kks3[2] # long time (11.85 s, 2013) -Kks3[3, 1, 1] - Kks3[3, 2] + Kks3[3, 2, 1] + Kks3[3, 3] sage: Kks3[3,1].coproduct() Kks3[] # Kks3[3, 1] - Kks3[1] # Kks3[2] + Kks3[1] # Kks3[2, 1] + 2*Kks3[1] # Kks3[3] + Kks3[1, 1] # Kks3[2] - Kks3[2] # Kks3[1] + Kks3[2] # Kks3[1, 1] + 2*Kks3[2] # Kks3[2] + Kks3[2, 1] # Kks3[1] + 2*Kks3[3] # Kks3[1] + Kks3[3, 1] # Kks3[]
Sage example in ./kschurnotes/notes-mike-anne.tex, line 5588::
sage: SymQ3 = Sym.kBoundedQuotient(3,t=1) sage: G1 = SymQ3.AffineGrothendieckPolynomial([1],6) sage: G2 = SymQ3.AffineGrothendieckPolynomial([2],6) sage: (G1*G2).lift().scalar(Kks3[3,1]) -1 """ |