# 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 2. `"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 (ex-fe8350)` ```magma > 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 -> SetIndx` ### `Coroots(R): RootSys -> SetIndx` ```magma 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 -> SetIndx` ### `PositiveCoroots(R): RootSys -> SetIndx` ```magma 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 -> SetIndx` ### `Coroot(R, r): RootSys, RngIntElt -> SetIndx` ```magma Basis: MonStgElt Default: "Standard" ``` The $r$th (co)root $\alpha_r$ (respectively, $\alpha_r^\star$) of the root system $R$. ### `RootPosition(R, v): RootSys, . -> SetIndx` ### `CorootPosition(R, v): RootSys, . -> SetIndx` ```magma 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 (ex-89c0a9)` ```magma > 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 > Root(R, 4); (1 -1 -1) > Root(R, 4 : Basis := "Root"); (2 1) > RootPosition(R, [1,-1,-1]); 4 > RootPosition(R, [2,1] : Basis := "Root"); 4 ``` ### `HighestRoot(R): RootSys -> .` ### `HighestCoroot(R): RootSys -> .` ```magma Basis: MonStgElt Default: "Standard" ``` The unique (co)root of greatest height in the irreducible root system $R$. ### `HighestLongRoot(R): RootSys -> .` ### `HighestLongCoroot(R): RootSys -> .` ```magma Basis: MonStgElt Default: "Standard" ``` The unique long (co)root of greatest height in the irreducible root system $R$. ### `HighestShortRoot(R): RootSys -> .` ### `HighestShortCoroot(R): RootSys -> .` ```magma Basis: MonStgElt Default: "Standard" ``` The unique short (co)root of greatest height in the irreducible root system $R$. ### `Example: Heighest Roots (ex-77bd29)` ```magma > R := RootSystem("G2"); > HighestRoot(R); (3 2) > HighestLongRoot(R); (3 2) > HighestShortRoot(R); (2 1) ``` ### `CoxeterForm(R): RootSys -> AlgMatElt` ### `DualCoxeterForm(R): RootSys -> AlgMatElt` ```magma 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 -> []` ```magma 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 -> []` ```magma 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 -> []` ```magma 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 (ex-a58034)` ```magma > 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 > perm eq &*[ ReflectionPermutation(R, r) : r in wd ]; true > > mx := CoreflectionMatrix(R, 4); > CorootPosition(R, Coroot(R,2) * mx) eq 2^perm; true ``` ## 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 $t = \hbox{\tt Sum(R,r,s)} \ne 0$ 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 (ex-8a578c)` ```magma > R := RootSystem("G2"); > Sum(R, 1, Negative(R,5)); 10 > IsPositive(R, 10); false > Negative(R, 10); 4 > P := PositiveRoots(R); > P[1] - P[5] eq -P[4]; true ``` ### `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 (ex-e2515b)` ```magma > R := RootSystem("G2"); > RootHeight(R, 5); 4 > F := CoxeterForm(R); > v := Root(R, 5); > (v*F, v) eq RootNorm(R, 5); true > IsLongRoot(R, 5); true > 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](../../references.md#cite-papi)]. ### `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 (ex-20f94b)` ```magma > R := RootSystem("A5"); > a := AdditiveOrder(R); > Position(a, 2); 6 > Position(a, 3); 10 > Position(a, Sum(R, 2, 3)); 7 ```