.. MODULEAUTHOR:: Nicolas M. Thiéry, Vic Reiner, ...
defQQ{mathbb{Q}}defCC{mathbb{C}}defHilb{operatorname{Hilb}}
This demonstration review some of the features of Coxeter and reflection groups (basic manipulations, related posets, calculation of Molien type series). It grew out of a live demo improvised with the participants during interactive sessions at the CRM workshop on reflection groups on May 29th of 2017.
Requirements
This demonstration requires gap3 to be installed.
sage: %display latex # not tested
Let's build the Coxeter group of type E_8, and do some sample calculations with it:
sage: W = CoxeterGroup(["E",8]) sage: W Finite Coxeter group over Integer Ring with Coxeter matrix: [1 2 3 2 2 2 2 2] [2 1 2 3 2 2 2 2] [3 2 1 3 2 2 2 2] [2 3 3 1 3 2 2 2] [2 2 2 3 1 3 2 2] [2 2 2 2 3 1 3 2] [2 2 2 2 2 3 1 3] [2 2 2 2 2 2 3 1] sage: W.cardinality() 696729600
By default, this Coxeter group is constructed as a matrix group:
sage: W.an_element() [ 0 1 0 0 0 0 0 -1] [ 0 0 1 0 0 0 0 -1] [ 1 1 0 0 0 0 0 -1] [ 0 1 1 0 0 0 0 -1] [ 0 0 0 1 0 0 0 -1] [ 0 0 0 0 1 0 0 -1] [ 0 0 0 0 0 1 0 -1] [ 0 0 0 0 0 0 1 -1]
Instead, it's possible to construct it as a permutation group, namely the group of permutation of its roots:
sage: W = CoxeterGroup(["E",8], implementation="permutation") sage: w = W.an_element(); w (1,121)(3,13)(9,22)(18,27)(19,23)(25,30)(29,35)(31,42)(34,40)(36,37)(38,45)(41,47)(43,49)(44,56)(48,51)(50,52)(54,59)(55,62)(57,64)(60,63)(68,73)(74,78)(96,98)(100,102)(101,106)(104,107)(108,110)(109,112)(111,113)(123,133)(129,142)(138,147)(139,143)(145,150)(149,155)(151,162)(154,160)(156,157)(158,165)(161,167)(163,169)(164,176)(168,171)(170,172)(174,179)(175,182)(177,184)(180,183)(188,193)(194,198)(216,218)(220,222)(221,226)(224,227)(228,230)(229,232)(231,233)
Finite and affine Coxeter groups can be specified as above using their classification, by providing their Cartan type. Here is a sample of all available Cartan types:
sage: CartanType.samples() [['A', 1], ['A', 5], ['B', 1], ['B', 5], ['C', 1], ['C', 5], ['D', 2], ['D', 3], ['D', 5], ['E', 6], ['E', 7], ['E', 8], ['F', 4], ['G', 2], ['I', 5], ['H', 3], ['H', 4], ['A', 1, 1], ... ['BC', 5, 2]^*]
It contains all exceptional types, and a couple representatives of each infinite families.
Let's play with elements and reduced words. One can construct an element from the Coxeter generators (also called simple reflections) with:
sage: s = W.simple_reflections() sage: w = s[1] * s[3] * s[2]; w (1,133,3)(2,122)(4,18,22,10,9,27)(13,123,121)(15,25,23,21,19,30)(17,36,35,28,29,37)(26,41,42,33,31,47)(32,48,45,39,38,51)(34,46,40)(43,53,49)(44,61,56)(50,58,52)(54,67,59)(55,66,62)(57,69,64)(60,70,63)(65,72)(68,77,73)(71,75)(74,83,78)(76,84)(79,81)(80,87)(82,88)(85,91)(86,90)(89,96,98)(92,94)(93,100,102)(95,99)(97,101,106)(103,104,107)(105,108,110)(109,115,112,111,114,113)(124,138,142,130,129,147)(135,145,143,141,139,150)(137,156,155,148,149,157)(146,161,162,153,151,167)(152,168,165,159,158,171)(154,166,160)(163,173,169)(164,181,176)(170,178,172)(174,187,179)(175,186,182)(177,189,184)(180,190,183)(185,192)(188,197,193)(191,195)(194,203,198)(196,204)(199,201)(200,207)(202,208)(205,211)(206,210)(209,216,218)(212,214)(213,220,222)(215,219)(217,221,226)(223,224,227)(225,228,230)(229,235,232,231,234,233)
Here is a short hand (note: the word need not be reduced):
sage: W.from_reduced_word([1,3,2]) (1,133,3)(2,122)(4,18,22,10,9,27)(13,123,121)(15,25,23,21,19,30)(17,36,35,28,29,37)(26,41,42,33,31,47)(32,48,45,39,38,51)(34,46,40)(43,53,49)(44,61,56)(50,58,52)(54,67,59)(55,66,62)(57,69,64)(60,70,63)(65,72)(68,77,73)(71,75)(74,83,78)(76,84)(79,81)(80,87)(82,88)(85,91)(86,90)(89,96,98)(92,94)(93,100,102)(95,99)(97,101,106)(103,104,107)(105,108,110)(109,115,112,111,114,113)(124,138,142,130,129,147)(135,145,143,141,139,150)(137,156,155,148,149,157)(146,161,162,153,151,167)(152,168,165,159,158,171)(154,166,160)(163,173,169)(164,181,176)(170,178,172)(174,187,179)(175,186,182)(177,189,184)(180,190,183)(185,192)(188,197,193)(191,195)(194,203,198)(196,204)(199,201)(200,207)(202,208)(205,211)(206,210)(209,216,218)(212,214)(213,220,222)(215,219)(217,221,226)(223,224,227)(225,228,230)(229,235,232,231,234,233) sage: w.reduced_word() [1, 2, 3] sage: w.reduced_words() [[1, 3, 2], [2, 1, 3], [1, 2, 3]]
The symmetric group is endowed with it's Coxeter (in fact Weyl) group structure:
sage: W = SymmetricGroup(3) sage: print(W.category()) Join of Category of finite enumerated permutation groups and Category of finite weyl groups
Some computations with the weak order:
sage: P = W.weak_lattice() sage: P.plot() Graphics object consisting of 13 graphics primitives
sage: s = W.simple_reflections() sage: P.le( s[1], s[2]*s[1]) False sage: P.join(s[1], s[2]*s[1]) (1,3)
and with the absolute poset of a reflection group:
sage: W = ReflectionGroup(6) sage: P = W.absolute_poset(); sage: P.plot() Graphics object consisting of 265 graphics primitives
Let's start by exploring the Shephard-Todd reflection group G_4:
sage: W = ReflectionGroup(4); W Irreducible complex reflection group of rank 2 and type ST4 sage: W.cardinality() 24 sage: W.is_isomorphic(SymmetricGroup(4)) False
It is constructed as a permutation group:
sage: w = W.an_element(); w (1,3,9)(2,4,7)(5,10,18)(6,11,16)(8,12,19)(13,15,20)(14,17,21)(22,23,24)
Here is how to recover the matrix action on V and V^*:
sage: m = w.to_matrix(); m
[ 1 0]
[ 0 E(3)]
sage: w.to_matrix("dual")
[ 1 0]
[ 0 E(3)^2]
Let's use Molien's formula to compute the Hilbert series H=Hilb(CC[V]^W,q) of the invariant ring CC[V]^W=S(V^*)^W:
sage: QQq = QQ['q'].fraction_field() sage: q = QQq.gen() sage: H = 1/W.cardinality() * sum( 1/det(1-q*w.to_matrix()) for w in W ); sage: H 1/(q^10 - q^6 - q^4 + 1)
We know that this should factor as frac{1}{prod 1-q^{d_i}}.
Frustrating as it is, Sage can't factor the above fraction as is:
sage: H.factor() Traceback (most recent call last): ... NotImplementedError
That's because it looks like a fraction in QQ(q) but it is in fact a fraction in the Universal Cyclotomic Field (the extension of QQ containing all roots of unity):
sage: H.parent() Fraction Field of Univariate Polynomial Ring in q over Universal Cyclotomic Field
To proceed, we first convert H into QQ(q):
sage: H = QQq(H) sage: H.parent() Fraction Field of Univariate Polynomial Ring in q over Rational Field
and then can finally factor it:
sage: factor(H.denominator()) (q - 1)^2 * (q + 1)^2 * (q^2 - q + 1) * (q^2 + 1) * (q^2 + q + 1)
This is a product of cyclotomic polynomials, and by manual inspection, one we can recover the desired form for the denominator of H:
sage: H.denominator() == (1-q^4)*(1-q^6) True
This is telling us that the invariant ring is generated by two invariants of degree 4 and 6. Let's double check this.
Sage can compute generators of an invariant ring of a finite matrix
group, but only over reasonably simple fields, which does not include
the Universal Cyclotomic Field. So we are going to convert our group
into a matrix group WM over the Cyclotomic Field of degree 3:
sage: K = CyclotomicField(3) sage: WM = MatrixGroup( [ matrix(K, w.to_matrix()) for w in W.gens()]) sage: WM Matrix group over Cyclotomic Field of order 3 and degree 2 with 2 generators ( [ 1 0] [2/3*zeta3 + 1/3 1/3*zeta3 - 1/3] [ 0 zeta3], [2/3*zeta3 - 2/3 1/3*zeta3 + 2/3] ) sage: WM.invariant_generators() [x1^4 - x1*x2^3, x1^6 + 5/2*x1^3*x2^3 - 1/8*x2^6]
We will use that V and V^* are irreducible representations together with the following relations between the Hilbert series of the corresponding isotypic components in the polynomial ring CC[V]^W with the exponents e_1,ldots,e_n and coexponents e_1^*,ldots,e_n^*:
\frac{1}{|W|} \sum_{w\in W} \frac{\chi_V(w)}{\det(1-qw)} = \Hilb(\CC[V]^W,q) \quad ( q^{e_1} + \cdots + q^{e_n})
\frac{1}{|W|} \sum_{w\in W} \frac{\chi_V^*(w)}{\det(1-qw)} = \Hilb(\CC[V]^W,q) \quad ( q^{e_1^*} + \cdots + q^{e_n^*})
sage: 1/W.cardinality() * sum( w.to_matrix().trace()/det(1-q*w.to_matrix()) for w in W ) / H
q^5 + q^3
sage: 1/W.cardinality() * sum( w.to_matrix("dual").trace()/det(1-q*w.to_matrix()) for w in W ) / H
q^3 + q
Let's do a consistency check with the degrees (which are the e_i+1) and the codegrees (which are the e_i^*-1):
sage: W.degrees() (4, 6) sage: W.codegrees() (2, 0)
Exercise
Compute the Hilbert series of (CC[V]otimes bigwedge^cdot V^*)^W via a Molien-type calculation:
\Hilb((\CC[V]\otimes \bigwedge^\cdot V^*)^W,q,t) = \frac{1}{|W|} \sum_{w\in W} \frac{\det(1+tw)}{\det(1-qw)}
and then compare it to the prediction of Solomon's formula, namely:
\Hilb((\CC[V]\otimes \bigwedge^\cdot V^*)^W,q,t) = \frac{\prod_{i=1}^n ( 1 + q^{e_i}t )}{\prod_{i=1}^n (1 - q^{d_i} )}
Solution
sage: QQqt = QQ['q,t'].fraction_field() sage: q,t = QQqt.gens() sage: Solomon = 1/W.cardinality() * sum( det(1+t*w.to_matrix()) / det(1-q*w.to_matrix()) for w in W ) sage: QQqt(Solomon) / H q^8*t^2 + q^5*t + q^3*t + 1 sage: _.factor() (q^3*t + 1) * (q^5*t + 1)