Roots and Coroots

Roots and Coroots#

The roots are stored as an indexed set

\[\{@\; \alpha_1,\dots,\alpha_N,\alpha_{N+1},\dots,\alpha_{2N} \; @\},\]

where \(\alpha_1,\dots,\alpha_N\) are the positive roots (in an order compatible with height), and \(\alpha_{N+1},\dots,\alpha_{2N}\) are the corresponding negative roots (i.e. \(\alpha_{i+N}=-\alpha_i\)). The simple roots are \(\alpha_1,\dots,\alpha_n\) where \(n\) is the rank.

Many of these functions have an optional argument Basis which may take one of the following values

1. "Standard": the standard basis for the (co)root space (this is the default); or

  1. "Root": the basis of simple (co)roots.

Accessing Roots and Coroots#

RootSpace(R): RootSys → ModTupFld#
CorootSpace(R): RootSys → ModTupFld#

The vector space containing the (co)roots of the root system \(R\), i.e. \(X\) (respectively, \(Y\)).

SimpleRoots(R): RootSys → Mtrx#
SimpleCoroots(R): RootSys → Mtrx#

The simple (co)roots of the root system \(R\) as the rows of a matrix, i.e. \(A\) (respectively, \(B\)).

Example: Root Space#
> R := RootSystem("G2");
> RootSpace(R);
Full Vector space of degree 2 over Rational Field
> CorootSpace(R);
Full Vector space of degree 2 over Rational Field
> SimpleRoots(R);
[1 0]
[0 1]
> SimpleCoroots(R);
[ 2 -3]
[-1  2]
> CartanMatrix(R);
[ 2 -1]
[-3  2]
NumberOfPositiveRoots(R): RootSys → RngIntElt#
NumPosRoots(R): RootSys → RngIntElt#

The number of positive roots of the root system \(R\). This is also the number of positive coroots. The total number of (co)roots is twice the number of positive (co)roots.

Roots(R): RootSys → $\{@@\}$#
Coroots(R): RootSys → $\{@@\}$#
Basis : MonStgElt                    Default: ``Standard"

The indexed set of (co)roots of the root system \(R\), i.e. \(\{@\,\alpha_1,\dots\alpha_{2N}\,@\}\) (respectively, \(\{@\,\alpha_1^\star,\dots\alpha_{2N}^\star\,@\}\)).

PositiveRoots(R): RootSys → $\{@@\}$#
PositiveCoroots(R): RootSys → $\{@@\}$#
Basis : MonStgElt                    Default: ``Standard"

The indexed set of positive (co)roots of the root system \(R\), i.e. \(\{@\,\alpha_1,\dots\alpha_N\,@\}\) (respectively, \(\{@\,\alpha_1^\star,\dots\alpha_N^\star\,@\}\)).

Root(R, r): RootSys, RngIntElt → $\{@@\}$#
Coroot(R, r): RootSys, RngIntElt → $\{@@\}$#
Basis : MonStgElt                    Default: ``Standard"

The \(r\)th (co)root \(\alpha_r\) (respectively, \(\alpha_r^\star\)) of the root system \(R\).

RootPosition(R, v): RootSys, . → $\{@@\}$#
CorootPosition(R, v): RootSys, . → $\{@@\}$#
Basis : MonStgElt                    Default: ``Standard"

If \(v\) is a (co)root in the root system \(R\), return its index; otherwise return 0. These functions will try to coerce \(v\), which can be a vector or a sequence representing a vector, into the appropriate vector space; \(v\) should be written with respect to the basis specified by the parameter Basis.

Example: Roots Coroots#
> A := Matrix(2,3, [1,-1,0, -1,1,-1]);
> B := Matrix(2,3, [1,-1,1, 0,1,-1]);
> R := RootSystem(A, B);
> Roots(R);
{@
    (1 -1  0),
    (-1  1 -1),
    (0  0 -1),
    (1 -1 -1),
    (2 -2 -1),
    (1 -1 -2),
    (-1  1  0),
    (1 -1  1),
    (0 0 1),
    (-1  1  1),
    (-2  2  1),
    (-1  1  2)
@}
> PositiveCoroots(R);
{@
    (1 -1  1),
    (0  1 -1),
    (1  2 -2),
    (2  1 -1),
    (1 0 0),
    (1  1 -1)
@}
> #Roots(R) eq 2*NumPosRoots(R);
true
%%a> assert $1;
> Root(R, 4);
(1 -1 -1)
> Root(R, 4 : Basis := "Root");
(2 1)
> RootPosition(R, [1,-1,-1]);
4
%%a> assert $1 eq 4;
> RootPosition(R, [2,1] : Basis := "Root");
4
%%a> assert $1 eq 4;
HighestRoot(R): RootSys → .#
HighestCoroot(R): RootSys → .#
Basis : MonStgElt                    Default: ``Standard"

The unique (co)root of greatest height in the irreducible root system \(R\).

HighestLongRoot(R): RootSys → .#
HighestLongCoroot(R): RootSys → .#
Basis : MonStgElt                    Default: ``Standard"

The unique long (co)root of greatest height in the irreducible root system \(R\).

HighestShortRoot(R): RootSys → .#
HighestShortCoroot(R): RootSys → .#
Basis : MonStgElt                    Default: ``Standard"

The unique short (co)root of greatest height in the irreducible root system \(R\).

Example: Heighest Roots#
> R := RootSystem("G2");
> HighestRoot(R);
(3 2)
> HighestLongRoot(R);
(3 2)
> HighestShortRoot(R);
(2 1)
CoxeterForm(R): RootSys → AlgMatElt#
DualCoxeterForm(R): RootSys → AlgMatElt#
Basis : MonStgElt                    Default: ``Standard"

The matrix of an inner product on the (co)root space of the root system \(R\) which is invariant under the action of the (co)roots. This inner product is uniquely determined up to a constant on each irreducible component of \(R\). The inner product is normalised so that the short roots in each crystallographic component have length one.

Reflections#

The root \(\alpha\) acts on the root space via the reflection \(s_\alpha\); the coroot \(\alpha^\star\) acts on the coroot space via the coreflection \(s_\alpha^\star\).

SimpleReflectionMatrices(R): RootSys → []#
SimpleCoreflectionMatrices(R): RootSys → []#
Basis : MonStgElt                    Default: ``Standard"

The sequence of matrices giving the action of the simple (co)roots of the root system \(R\) on the (co)root space, i.e. the matrices of \(s_{\alpha_1},\dots,s_{\alpha_n}\) (respectively, \(s_{\alpha_1}^\star,\dots,s_{\alpha_n}^\star\)).

ReflectionMatrices(R): RootSys → []#
CoreflectionMatrices(R): RootSys → []#
Basis : MonStgElt                    Default: ``Standard"

The sequence of matrices giving the action of the (co)roots of the root system \(R\) on the (co)root space, i.e. the matrices of \(s_{\alpha_1},\dots,s_{\alpha_{2N}}\) (respectively, \(s_{\alpha_1}^\star,\dots,s_{\alpha_{2N}}^\star\)).

ReflectionMatrix(R, r): RootSys, RngIntElt → []#
CoreflectionMatrix(R, r): RootSys, RngIntElt → []#
Basis : MonStgElt                    Default: ``Standard"

The matrix giving the action of the \(r\)th (co)root of the root system \(R\) on the (co)root space, i.e. the matrix of \(s_{\alpha_r}\) (respectively, \(s_{\alpha_r}^\star\)).

SimpleReflectionPermutations(R): RootSys → []#

The sequence of permutations giving the action of the simple (co)roots of the root system \(R\) on the (co)roots. This action is the same for roots and coroots.

ReflectionPermutations(R): RootSys → []#

The sequence of permutations giving the action of the (co)roots of the root system \(R\) on the (co)roots. This action is the same for roots and coroots.

ReflectionPermutation(R, r): RootSys, RngIntElt → []#

The permutation giving the action of the \(r\)th (co)root of the root system \(R\) on the (co)roots. This action is the same for roots and coroots.

ReflectionWords(R): RootSys → []#

The sequence of words in the simple reflections for all the reflections of the root system \(R\). These words are given as sequences of integers. In other words, if \([a_1,\dots,a_l] = \hbox{\tt ReflectionWords(R)[r]}\), then \(s_{\alpha_r} = s_{\alpha_{a_1}} \cdots s_{\alpha_{a_l}}\).

ReflectionWord(R, r): RootSys, RngIntElt → []#

The word in the simple reflections for the \(r\)th reflection of the root system \(R\). The word is given as a sequence of integers. In other words, if \([a_1,\dots,a_l] = \hbox{\tt ReflectionWord(R,r)}\), then \(s_{\alpha_r} = s_{\alpha_{a_1}} \cdots s_{\alpha_{a_l}}\).

Example: Action#
> R := RootSystem("B3");
> mx := ReflectionMatrix(R, 4);
> perm := ReflectionPermutation(R, 4);
> wd := ReflectionWord(R, 4);
> RootPosition(R, Root(R,2) * mx) eq 2^perm;
true
%%a> assert $1;
> perm eq &*[ ReflectionPermutation(R, r) : r in wd ];
true
%%a> assert $1;
>
> mx := CoreflectionMatrix(R, 4);
> CorootPosition(R, Coroot(R,2) * mx) eq 2^perm;
true
%%a> assert $1;

Operations and Properties for Roots and Coroot Indices#

Sum(R, r, s): RootSys, RngIntElt, RngIntElt → RngIntElt#

The index of the sum of the \(r\)th and \(s\)th roots in the crystallographic root system \(R\), or 0 if the sum is not a root. In other words, if \(t=\hbox{\tt Sum(R,$r$,$s$)}\ne0\) then \(\alpha_t=\alpha_r+\alpha_s\). We require \(\alpha_r\ne\pm\alpha_s\).

IsPositive(R, r): RootSys, RngIntElt → BoolElt#

Returns true if, and only if, the \(r\)th (co)root of the root system \(R\) is a positive root.

IsNegative(R, r): RootSys, RngIntElt → BoolElt#

Returns true if, and only if, the \(r\)th (co)root of the root system \(R\) is a negative root.

Negative(R, r): RootSys, RngIntElt → RngIntElt#

The index of the negative of the \(r\)th (co)root of the root system \(R\). In other words, if \(s = \hbox{\tt Negative(R,$r$)}\) then \(\alpha_s=-\alpha_r\).

Example: Root Arithmetic#
> R := RootSystem("G2");
> Sum(R, 1, Negative(R,5));
10
%%a> assert $1 eq 10;
> IsPositive(R, 10);
false
%%a> assert not $1;
> Negative(R, 10);
4
%%a> assert $1 eq 4;
> P := PositiveRoots(R);
> P[1] - P[5] eq -P[4];
true
%%a> assert $1;
RootHeight(R, r): RootSys, RngIntElt → RngIntElt#
CorootHeight(R, r): RootSys, RngIntElt → RngIntElt#

The height of the \(r\)th (co)root of the root system \(R\), i.e. the sum of the coefficients of \(\alpha_r\) (respectively, \(\alpha_r^\star\)) with respect to the simple (co)roots.

RootNorms(R): RootSys → [RngIntElt]#
CorootNorms(R): RootSys → [RngIntElt]#

The sequence of squares of the lengths of the (co)roots of the root system \(R\).

RootNorm(R, r): RootSys, RngIntElt → RngIntElt#
CorootNorm(R, r): RootSys, RngIntElt → RngIntElt#

The square of the length of the \(r\)th (co)root of the root system \(R\).

IsLongRoot(R, r): RootSys, RngIntElt → BoolElt#

Returns true if, and only if, the \(r\)th root of the root system \(R\) is long. This only makes sense for irreducible crystallographic root systems. Note that for non-reduced root systems, the roots which are not indivisible are actually longer than the long ones.

IsShortRoot(R, r): RootSys, RngIntElt → BoolElt#

Returns true if, and only if, the \(r\)th root of the root system \(R\) is short. This only makes sense for irreducible crystallographic root systems.

IsIndivisibleRoot(R, r): RootSys, RngIntElt → BoolElt#

Returns true if, and only if, the \(r\)th root of the root system \(R\) is indivisible, ie, \(\alpha_r/2\) is not a root.

LeftString(R, r, s): RootSys, RngIntElt, RngIntElt → RngIntElt#

Indices in the crystallographic root system \(R\) of the left string through \(\alpha_s\) in the direction of \(\alpha_r\), i.e. the indices of \(\alpha_s-\alpha_r,\alpha_s-2\alpha_r,\dots,\alpha_s-p\alpha_r\). In other words, this returns the sequence \([r_1,\dots,r_p]\) where \(\alpha_{r_i}=\alpha_s-i\alpha_r\) and \(\alpha_s-(p+1)\alpha_r\) is not a root. We require that \(\alpha_r\ne\pm\alpha_s\).

RightString(R, r, s): RootSys, RngIntElt, RngIntElt → RngIntElt#

Indices in the crystallographic root system \(R\) of the left string through \(\alpha_s\) in the direction of \(\alpha_r\), i.e. the indices of \(\alpha_s+\alpha_r,\alpha_s+2\alpha_r,\dots,\alpha_s+q\alpha_r\). In other words, this returns the sequence \([r_1,\dots,r_q]\) where \(\alpha_{r_i}=\alpha_s+i\alpha_r\) and \(\alpha_s+(q+1)\alpha_r\) is not a root. We require that \(\alpha_r\ne\pm\alpha_s\).

LeftStringLength(R, r, s): RootSys, RngIntElt, RngIntElt → RngIntElt#

The largest \(p\) such that \(\alpha_s-p\alpha_r\) is a root. We require that the root system \(R\) be crystallographic and \(\alpha_s\ne\pm\alpha_r\).

RightStringLength(R, r, s): RootSys, RngIntElt, RngIntElt → RngIntElt#

The largest \(q\) such that \(\alpha_s+q\alpha_r\) is a root. We require that the root system \(R\) be crystallographic and \(\alpha_s\ne\pm\alpha_r\).

Example: Root Operations#
> R := RootSystem("G2");
> RootHeight(R, 5);
4
%%a> assert $1 eq 4;
> F := CoxeterForm(R);
> v := Root(R, 5);
> (v*F, v) eq RootNorm(R, 5);
true
%%a> assert $1;
> IsLongRoot(R, 5);
true
%%a> assert $1;
> LeftString(R, 1, 5);
[ 4, 3, 2 ]
> roots := Roots(R);
> for i in [1..3] do
>   RootPosition(R, roots[5]-i*roots[1]);
> end for;
4
3
2
> R := RootSystem("BC2");
> Root(R,2), IsIndivisibleRoot(R,2);
(0 1) true
> Root(R,4), IsIndivisibleRoot(R,4);
(0 2) false
AdditiveOrder(R): RootSys → SeqEnum#

An additive order on the positive roots of the root system \(R\), i.e. a sequence containing the numbers \(1,\dots,N\) in some order so that \(\alpha_r+\alpha_s=\alpha_t\) implies \(t\) is between \(r\) and \(s\). This is computed using the techniques of [Papi, 1994].

IsAdditiveOrder(R, Q): RootSys, [RngIntElt] → BoolElt#

Returns true if, and only if, the sequence \(Q\) gives an additive order on a set of positive roots of the root system \(R\). \(Q\) must be a sequence of integers in the range \([1..N]\), where \(N\) is the number of positive roots of \(R\), with no gaps or repeats.

Example: Additive Order#
> R := RootSystem("A5");
> a := AdditiveOrder(R);
> Position(a, 2);
6
%%a> assert $1 eq 6;
> Position(a, 3);
10
%%a> assert $1 eq 10;
> Position(a, Sum(R, 2, 3));
7
%%a> assert $1 eq 7;