This document is one of More SageMath Tutorials. You may edit it on github. $$\def\NN{\mathbb{N}}$$ $$\def\ZZ{\mathbb{Z}}$$ $$\def\QQ{\mathbb{Q}}$$ $$\def\RR{\mathbb{R}}$$ $$\def\CC{\mathbb{C}}$$

# Demonstration: Algebraic Combinatorics¶

## Tableaux and the like¶

sage: s = Permutation([5,3,2,6,4,8,9,7,1])
sage: s

sage: (p,q) = s.robinson_schensted()
sage: p.pp()
1  4  7  9
2  6  8
3
5

sage: p.row_stabilizer()
Permutation Group with generators [(), (7,9), (6,8), (4,7), (2,6), (1,4)]


## Symmetric functions¶

Usual bases:

sage: Sym = SymmetricFunctions(QQ); Sym
Symmetric Functions over Rational Field
sage: Sym.inject_shorthands()

sage: m(( ( h[2,1] * ( 1 + 3 * p[2,1]) ) + s(s)))


Macdonald polynomials:

sage: J = MacdonaldPolynomialsJ(QQ)
sage: P = MacdonaldPolynomialsP(QQ)
sage: Q = MacdonaldPolynomialsQ(QQ)
sage: J
Macdonald polynomials in the J basis over Fraction Field of Multivariate Polynomial Ring in q, t over Rational Field
sage: P(J[2,2] + 3 * Q[3,1])


## Root systems¶

sage: L = RootSystem(['A',2,1]).weight_space()
sage: L.plot(size=[[-1..1],[-1..1]],alcovewalks=[[0,2,0,1,2,1,2,0,2,1]])

sage: W = WeylGroup(["B", 3])
sage: W.cayley_graph(side = "left").plot3d(color_by_label = True)


## GAP at work¶

sage: print(W.character_table())  # Thanks GAP!
CT1

2  4  4  3  3  4  3  1  1  3  4
3  1  .  .  .  .  .  1  1  .  1

1a 2a 2b 4a 2c 2d 6a 3a 4b 2e

X.1      1  1  1  1  1  1  1  1  1  1
X.2      1  1  1 -1 -1 -1 -1  1  1 -1
X.3      1  1 -1 -1  1 -1  1  1 -1  1
X.4      1  1 -1  1 -1  1 -1  1 -1 -1
X.5      2  2  .  . -2  .  1 -1  . -2
X.6      2  2  .  .  2  . -1 -1  .  2
X.7      3 -1  1  1  1 -1  .  . -1 -3
X.8      3 -1 -1 -1  1  1  .  .  1 -3
X.9      3 -1 -1  1 -1 -1  .  .  1  3
X.10     3 -1  1 -1 -1  1  .  . -1  3

sage: type(W.character_table())

sage: G = gap(W); G

sage: G.Ch

sage: T = G.CharacterTable(); T

sage: T.Irr()[10,10]

sage: type(T.Irr()[10,10])


## Coxeter3 at work¶

sage: W3 = CoxeterGroup(W, implementation="coxeter3")
sage: KL = matrix([ [ W3.kazhdan_lusztig_polynomial(u,v) if u.bruhat_le(v) else 0 for u in W3 ]
....:             for v in W3])
sage: show(KL)

sage: W = WeylGroup(["C", 3, 1])
sage: W
Weyl Group of type ['C', 3, 1] (as a matrix group acting on the root space)

sage: W.from_reduced_word([1,2,3,0,3,0,3,2,1,3,3,2]).stanley_symmetric_function()
256*m[1, 1, 1, 1, 1, 1] + 128*m[2, 1, 1, 1, 1] + 64*m[2, 2, 1, 1] + 32*m[2, 2, 2] + 48*m[3, 1, 1, 1] + 24*m[3, 2, 1] + 8*m[3, 3] + 16*m[4, 1, 1] + 8*m[4, 2] + 4*m[5, 1]


## Crystals¶

sage: latex.jsmath_avoid_list(['tikzpicture'])
sage: K = KirillovReshetikhinCrystal(['A',3,1], 2,2)
sage: G = K.digraph()
sage: G.set_latex_options(format = "dot2tex", edge_labels = True, color_by_label = {0:"black", 1:"blue", 2:"red", 3:"green"}, edge_options = lambda (u,v,label):({"backward":label ==0}))
sage: view(G, viewer="pdf", tightpage=True)