Exploring posets related to coxeter groups

%display latex

W=CoxeterGroup(["B",3])

P = W.bruhat_poset()

G = P.hasse_diagram()

G.set_latex_options(format="dot2tex")

view(G)

W=CoxeterGroup(["B",4])

c=W.coxeter_element()

c.absolute_covers()

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left[\right]$$

c.descents()

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left[4\right]$$

c.matrix()

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} 0 & 0 & 1 & -\sqrt{2} \\ 1 & 0 & 1 & -\sqrt{2} \\ 0 & 1 & 1 & -\sqrt{2} \\ 0 & 0 & \sqrt{2} & -1 \end{array}\right)$$

c.weak_covers()

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(\begin{array}{rrrr} 0 & 0 & -1 & \sqrt{2} \\ 1 & 0 & -1 & \sqrt{2} \\ 0 & 1 & -1 & \sqrt{2} \\ 0 & 0 & 0 & 1 \end{array}\right)\right]$$

L=W.weak_lattice()

print L.category()

Category of finite enumerated lattice posets

L.is_supersolvable()

$$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{False}$$

L.is_selfdual()

$$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}$$

L.chains()

$$\newcommand{\Bold}[1]{\mathbf{#1}}\verb|Set|\phantom{\verb!x!}\verb|of|\phantom{\verb!x!}\verb|chains|\phantom{\verb!x!}\verb|of|\phantom{\verb!x!}\verb|Finite|\phantom{\verb!x!}\verb|lattice|\phantom{\verb!x!}\verb|containing|\phantom{\verb!x!}\verb|384|\phantom{\verb!x!}\verb|elements|$$

The absolute order is not yet available for Weyl groups. In the mean time, we construct the same group as a reflection group.

W = ReflectionGroup([2,1,3])

W.cardinality()

$$\newcommand{\Bold}[1]{\mathbf{#1}}48$$

P = W.absolute_poset()

s = W.simple_reflections()

s

$$\newcommand{\Bold}[1]{\mathbf{#1}}\verb|Finite|\phantom{\verb!x!}\verb|family|\phantom{\verb!x!}\verb|{1:|\phantom{\verb!x!}\verb|(1,10)(2,6)(5,8)(11,15)(14,17),|\phantom{\verb!x!}\verb|2:|\phantom{\verb!x!}\verb|(1,4)(2,11)(3,5)(8,9)(10,13)(12,14)(17,18),|\phantom{\verb!x!}\verb|3:|\phantom{\verb!x!}\verb|(2,5)(3,12)(4,7)(6,8)(11,14)(13,16)(15,17)}|$$

s[1]

$$\newcommand{\Bold}[1]{\mathbf{#1}}(1,10)(2,6)(5,8)(11,15)(14,17)$$

I = P.interval(W.one(), s[1]*s[2]*s[3])

I

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left[, (3,18)(5,17)(7,16)(8,14)(9,12), (2,5)(3,12)(4,7)(6,8)(11,14)(13,16)(15,17), (2,15)(3,9)(4,13)(6,11)(12,18), (2,17)(4,16)(5,15)(6,14)(7,13)(8,11)(9,18), (2,17,15,5)(3,9,12,18)(4,16,13,7)(6,14,11,8), (1,4)(2,11)(3,5)(8,9)(10,13)(12,14)(17,18), (1,4)(2,11)(3,17)(5,18)(7,16)(8,12)(9,14)(10,13), (1,7)(2,12)(3,11)(5,14)(6,9)(10,16)(15,18), (1,7,4)(2,14,3)(5,12,11)(6,8,9)(10,16,13)(15,17,18), (1,10)(2,6)(5,8)(11,15)(14,17), (1,10)(2,8)(3,12)(4,7)(5,6)(11,17)(13,16)(14,15), (1,13)(3,8)(4,10)(5,9)(6,15)(12,17)(14,18), (1,13,10,4)(2,6,11,15)(3,5,9,8)(12,14,18,17), (1,16)(2,9)(3,15)(6,12)(7,10)(8,17)(11,18), (1,16)(2,3)(4,13)(6,18)(7,10)(8,17)(9,15)(11,12), (1,16,13)(2,5,9)(3,6,17)(4,10,7)(8,12,15)(11,14,18), (1,16,4)(2,8,18)(3,15,5)(6,14,12)(7,13,10)(9,11,17), (1,16,10,7)(2,9,6,12)(3,11,18,15)(5,8,14,17), (1,16,13,10,7,4)(2,8,12,11,17,3)(5,9,6,14,18,15)\right]$$

I = P.subposet(I)

I.plot()

I.is_lattice()

$$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}$$

I = LatticePoset(I)

I.is_supersolvable()

$$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}$$