Root systems

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#Root systems

A root system $\Phi \subset V$ is a set of vectors in $V$ such that

$\Phi$ spans $V,$
only $x$ and $-x$ are the scalar multiples of $x$ in $\Phi,$
$\Phi$ is closed under reflection in the hyperplane defined by $x,$ for every $x \in \Phi,$
for $x, y \in \Phi,$ $x$ projected onto $y$ is a half-integral multiple of $y.$

The root lattice is the lattice in $V$ generated by $\Phi.$ That is, its points are integer linear combinations of roots.

A simple root is a root that cannot be written as a positive sum of other roots. Each root $\alpha$ defines a hyperplane, where $\pm \alpha$ are normals to this hyperplane. These hyperplanes divide the space into Weyl chambers. By picking a vector $v$ in the interior of a Weyl chamber, we define a half-space. The positive roots $\Phi^+$ are the roots in this half-space and the base (with respect to $\Phi^+$) is the set of simple roots in $\Phi^+.$

Given $\Phi^+,$ the root poset is a graded partial order on $\Phi^+,$ where $\alpha \geq \beta$ if $\alpha - \beta$ is a sum of simple positive roots. The grading is given by the number of simple positive roots needed to express the root as a sum.

Example (Type $A_2$).

For type $A_2,$ take \[\alpha_1=e_1-e_2,\qquad \alpha_2=e_2-e_3\] as simple roots. The positive roots are \[\Phi^+=\{\alpha_1,\alpha_2,\alpha_1+\alpha_2\}.\] In the root poset, the highest root $\alpha_1+\alpha_2$ covers both $\alpha_1$ and $\alpha_2.$ The Weyl group is $\symS_3.$

The coroot of a root is defined as $\alpha^\vee = 2\alpha/(\alpha,\alpha).$ The set of coroots defines the dual or inverse root system. A root system and its dual have the same Weyl group.

#Classification of simple root systems

The simple root systems are classified by the Dynkin diagrams, $A_{n \geq 1},$ $B_{n \geq 2},$ $C_{n \geq 3},$ $D_{n \geq 4},$ $E_6,$ $E_7,$ $E_8,$ $F_4$ and $G_2.$

#Weyl groups and Lie groups

$ \text{Root system} $ $ \text{Weyl group} $ $ \text{Lie group} $ $ A_{n-1} $ $ \symS_n $ $ \mathrm{SL}_{n}(\setC) $ $ B_n $ $ \symB_n $ $ \mathrm{SO}_{2n+1}(\setC) $ $ C_n $ $ \symB_n $ $ \mathrm{Sp}_{2n}(\setC) $ $ D_n $ $ \symD_n $ $ \mathrm{SO}_{2n}(\setC) $

The Weyl group $\symS_n$ is the permutation group, which we can think of as $n{\times}n$ permutation matrices.

The group $\symB_n$ is known as the hyperoctahedral group. It can be seen as the group of signed permutations: the $n{\times}n$ permutation matrices with nonzero entries allowed to be $\pm 1.$ This can be thought of as the group of permutations of type $B$. The group $\symD_n \subset \symB_n$ is the subgroup of the signed permutation matrices where the number of negative entries is even.

The groups $\symS_n,$ $\symB_n$ and $\symD_n$ act on vectors via matrix multiplication.

For a compact background discussion of root systems and Weyl groups in the classical Schubert-polynomial setting, see [BH95].

Finite real reflection groups are finite subgroups of $\mathrm{GL}(V)$ generated by reflections. Choosing normal vectors to the reflecting hyperplanes gives a root-system viewpoint on the group, while the crystallographic integrality condition is the additional restriction that leads to Weyl groups in Lie theory. This distinction is visible in type $B/C:$ the crystallographic root systems $B_n$ and $C_n$ are different, but they determine the same reflection group [Stu08].

Let $W$ be a reflection group. The ring of polynomials in $\setR[x_1,\dotsc,x_n]$ invariant under $W$ can be spanned by homogeneous polynomials. The degrees of these polynomials only depend on $W.$ The order of the reflection group is the product of the degrees.

$ \text{Reflection group $W$} $ $ \text{Degrees} $ $ \text{Order}$ $ A_{n-1} $ $ 2,3,\dotsc,n $ $ n!$ $ B_n/C_n $ $ 2,4,\dotsc,2n $ $ 2^n n!$ $ D_n $ $ 2,4,\dotsc,2(n-1),n $ $ 2^{n-1} n! $

The exponents of a finite Coxeter group are the degrees minus one. For the classical Weyl groups above, the exponent multisets are $ \text{Type} $ $ \text{Exponents}$ $ A_{n-1} $ $ 1,2,\dotsc,n-1$ $ B_n/C_n $ $ 1,3,5,\dotsc,2n-1$ $ D_n $ $ 1,3,5,\dotsc,2n-3,n-1$ These exponents enter the definition of the generalized Catalan numbers of finite Coxeter groups. For identities involving power sums of Coxeter exponents, see [BS12].

For the broader Coxeter-group viewpoint and a translation table between the classical types, see the Coxeter groups and types page.

Bibliography

[BH95]Sara Billey and Mark Haiman. Schubert polynomials for the classical groups. Journal of the American Mathematical Society, 8(2):443–482, 1995.

.bib

@article{BilleyHaiman1995,
  doi = {10.1090/s0894-0347-1995-1290232-1},
  url2 = {https://doi.org/10.1090/s0894-0347-1995-1290232-1},
  year = {1995},
  publisher = {American Mathematical Society ({AMS})},
  volume = {8},
  number = {2},
  pages = {443--482},
  author = {Sara Billey and Mark Haiman},
  title = {Schubert polynomials for the classical groups},
  journal = {Journal of the American Mathematical Society}
}

[BS12]John M. Burns and Ruedi Suter. Power sums of Coxeter exponents. Advances in Mathematics, 231(3–4):1291–1307, 2012.

.bib

@article{BurnsSuter2012,
  author = {John M. Burns and Ruedi Suter},
  title = {Power sums of {C}oxeter exponents},
  year = {2012},
  journal = {Advances in Mathematics},
  volume = {231},
  number = {3--4},
  pages = {1291--1307},
  publisher = {Elsevier BV},
  doi = {10.1016/j.aim.2012.06.020},
  url = {https://doi.org/10.1016/j.aim.2012.06.020}
}

[Stu08]Christian Stump. $q,t$-Fuß–Catalan numbers for finite reflection groups. Universität Wien, 2008.

.bib

@phdthesis{Stump2008Thesis,
  author = {Christian Stump},
  title = {$q,t$-{F}u{\ss}--{C}atalan numbers for finite reflection groups},
  school = {Universit{\"a}t Wien},
  year = {2008},
  url = {https://www.mat.univie.ac.at/~schlosse/pdf/ChristianStumpDissertation.pdf}
}