# Roots, Coroots and Weights 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. 2. `"Root"`: the basis of simple (co)roots. 3. `"Weight"`: the basis of fundamental (co)weights (see Subsection [Weights](#subsectgrplierootweight) below). ## Accessing Roots and Coroots ### `RootSpace(G): GrpLie -> Lat` ### `CorootSpace(G): GrpLie -> Lat` The lattice containing the (co)roots of the group of Lie type $G$. ### `SimpleRoots(G): GrpLie -> Mtrx` ### `SimpleCoroots(G): GrpLie -> Mtrx` The simple (co)roots of the group of Lie type $G$ as the rows of a matrix. ### `NumberOfPositiveRoots(G): GrpLie -> RngIntElt` ### `NumPosRoots(G): GrpLie -> RngIntElt` The number of positive roots of the group of Lie type $G$. ### `Roots(G): GrpLie -> SetIndx` ### `Coroots(G): GrpLie -> SetIndx` ```magma Basis: MonStgElt Default: "Standard" ``` An indexed set containing the (co)roots of the group of Lie type $G$. ### `PositiveRoots(G): GrpLie -> SetIndx` ### `PositiveCoroots(G): GrpLie -> SetIndx` ```magma Basis: MonStgElt Default: "Standard" ``` An indexed set containing the positive (co)roots of the group of Lie type $G$. ### `Root(G, r): GrpLie, RngIntElt -> SetIndx` ### `Coroot(G, r): GrpLie, RngIntElt -> SetIndx` ```magma Basis: MonStgElt Default: "Standard" ``` The $r$th (co)root of the group of Lie type $G$. ### `RootPosition(G, v): GrpLie, . -> SetIndx` ### `CorootPosition(G, v): GrpLie, . -> SetIndx` ```magma Basis: MonStgElt Default: "Standard" ``` If $v$ is a (co)root of the group of Lie type $G$, this returns its position; otherwise it returns 0. ### `Example: Roots Coroots (ex-ac3db4)` ```magma > G := GroupOfLieType("A3", 25 : Isogeny := 2); > Roots(G); {@ (1 0 0), (0 1 0), (1 0 2), (1 1 0), (1 1 2), (2 1 2), (-1 0 0), (0 -1 0), (-1 0 -2), (-1 -1 0), (-1 -1 -2), (-2 -1 -2) @} > PositiveCoroots(G); {@ (2 -1 -1), (-1 2 0), (0 -1 1), (1 1 -1), (-1 1 1), (1 0 0) @} > #Roots(G) eq 2*NumPosRoots(G); true > Coroot(G, 4); (1 1 -1) > Coroot(G, 4 : Basis := "Root"); (1 1 0) > CorootPosition(G, [1,1,-1]); 4 > CorootPosition(G, [1,1,0] : Basis := "Root"); 4 ``` ### `HighestRoot(G): GrpLie -> LatElt` ### `HighestLongRoot(G): GrpLie -> LatElt` ```magma Basis: MonStgElt Default: "Standard" ``` The unique (long) root of greatest height in the root datum of the group of Lie type $G$. ### `HighestShortRoot(G): GrpLie -> LatElt` ```magma Basis: MonStgElt Default: "Standard" ``` The unique short root of greatest height in the root datum of the group of Lie type $G$. ### `Example: Heighest Roots (ex-69e0c7)` ```magma > G := GroupOfLieType("G2", RealField()); > HighestRoot(G); (3 2) > HighestLongRoot(G); (3 2) > HighestShortRoot(G); (2 1) ``` ## Reflections The reflections in the Weyl group have representatives in the group of Lie type. ### `Reflections(G): GrpLie -> GrpLieElt` The sequence of representatives of reflections in the group of Lie type $G$. ### `Reflection(G, r): GrpLie, RngIntElt -> GrpLieElt` The representative of the reflections in the $r$th root in the group of Lie type $G$. ### `Example: Reflections (ex-ff84ed)` ```magma > G := GroupOfLieType("A2", Rationals()); > Reflections(G); [ n1 , n2 , n1 n2 n1 ] ``` ## Operations and Properties for Root and Coroot Indices ### `RootHeight(G, r): GrpLie, RngIntElt -> RngIntElt` ### `CorootHeight(G, r): GrpLie, RngIntElt -> RngIntElt` The height of the $r$th (co)root of the group of Lie type $G$, i.e. the sum of the coefficients of $\alpha_r$ (resp. $\alpha_r^\star$) with respect to the simple (co)roots. ### `RootNorms(G): GrpLie -> [RngIntElt]` ### `CorootNorms(G): GrpLie -> [RngIntElt]` The sequence of squares of the lengths of the (co)roots of the group of Lie type $G$. ### `RootNorm(G, r): GrpLie, RngIntElt -> RngIntElt` ### `CorootNorm(G, r): GrpLie, RngIntElt -> RngIntElt` The square of the length of the $r$th (co)root of the group of Lie type $G$. ### `IsLongRoot(G, r): GrpLie, RngIntElt -> BoolElt` Returns `true` if, and only if, the $r$th root of the group of Lie type $G$ is long, i.e. the $r$th coroot is short. ### `IsShortRoot(G, r): GrpLie, RngIntElt -> BoolElt` Returns `true` if, and only if, the $r$th root of the group of Lie type $G$ is short, i.e. the $r$th coroot is long. ### `AdditiveOrder(G): GrpLie -> SeqEnum` The additive order on the positive roots of the group of Lie type $G$ equal to the Papi order of the longest word $w_0$ of the Weyl group of $G$; it corresponds to the order of roots in a reduced expression for $w_0$. If $\alpha_r$, $\alpha_s$ and $\alpha_t$ are positive roots and $\alpha_r+\alpha_s=\alpha_t$, then $t$ lies between $r$ and $s$. It is computed using the techniques of [[Papi, 1994](../../references.md#cite-papi)]. ### `Example: Additive Order (ex-dabc7d)` ```magma > G := GroupOfLieType("A5", GF(3)); > a := AdditiveOrder(G); > Position(a, 2); 6 > Position(a, 3); 10 ``` ## Weights ### `WeightLattice(G): GrpLie -> Lat` ### `CoweightLattice(G): GrpLie -> Lat` The (co)weight lattice of the group of Lie type $G$. ### `FundamentalWeights(G): GrpLie -> Mtrx` ### `FundamentalCoweights(G): GrpLie -> Mtrx` ```magma Basis: MonStgElt Default: "Standard" ``` The fundamental (co)weights of the group of Lie type $G$ as the rows of a matrix. ### `DominantWeight(G, v): GrpLie, . -> ModTupFldElt, GrpFPCoxElt` ```magma Basis: MonStgElt Default: "Standard" ``` The unique dominant weight in the same $W$-orbit as $v$, where $W$ is the Weyl group of $G$ and $v$ is a weight given as a vector or a sequence representing a vector. The second value returned is a Weyl group element taking $v$ to the dominant weight.