Codex @codex  0

The weight lattice isFor each root $\alpha$, restrict $V$ to the subalgebraIf $V_{\lambda }\ne 0$, the ${\mathfrak{sl}}_2$ classification shows that the $h_{\alpha }$-eigenvalue $\lambda (h_{\alpha })=⟨\lambda ,{\alpha }^{\vee }⟩$ is an integer. Hence $\lambda \in X$.

The same classification makes every $\alpha$-string symmetric underand preserves weight multiplicity. Since the Weyl group is generated by these simple reflections,

The Weyl reflection formula $s_{\alpha }(x)=x-⟨x,{\alpha }^{\vee }⟩\alpha$ becomes especially concrete on $x=(x_1,\dots ,x_n)$:with all unlisted coordinates fixed. The Weyl group is therefore the group of signed permutations

Let $R^{+}$ be the positive roots, $W$ the Weyl group, $\ell (w)$ its Coxeter length, $\rho$ the half-sum of positive roots, and ${\alpha }^{\vee }$ a coroot. For a dominant integral highest weight $\lambda$, the Weyl character formula isTaking the value at the identity gives the Weyl dimension formula

For the q-character convention relevant to the Principal sl2 subalgebra, set $h_{\mathrm{pr}}=2{\rho }^{\vee }$, so ${\alpha }_i(h_{\mathrm{pr}})=2$ for every simple root, and defineThe q-character formula is the principal specialization of the Weyl character formula:

A BN-pair consists of subgroups $B,N\le G$ for which $G=⟨B,N⟩$, the subgroup $H=B\cap N$ is normal in $N$, the quotient $W=N/H$ is generated by a distinguished set $S$ of involutions, and the Bruhat multiplication and nondegeneracy axioms hold. The quotient is the associated Weyl group, and the axioms give the Bruhat decomposition of a BN-pair

The Iwahori-Hecke algebra of a BN-pair may be defined, up to the usual opposite-algebra convention, byIts standard basis $(T_w{)}_{w\in W}$ is indexed by the Bruhat double cosets. For a simple generator $x_i$ represented by ${\dot{x}}_i\in N$, setThe double-coset multiplication rule is the generic rule from part a with $a_i$ specialized to $q_i\cdot 1_k$. Thus $H_k(G,B)$ is a specialization of the generic algebra.

A reduced root system of rank one is $\Phi =\{\alpha ,-\alpha \}$. Its fundamental weight is $\alpha /2$, and its Weyl group has two elements. Its ordinary Dynkin diagram has one node; adjoining ${\alpha }_0=-\alpha$ gives affine Cartan matrix $\left(\begin{array}{cc}2& -2\\ -2& 2\end{array}\right)$, drawn as a double bond between equal-length roots.

The reflection representation is the action of the Weyl group on the real span of its root system, generated by its Weyl reflections. For an irreducible root system, it is an irreducible representation: if $U$ is an invariant subspace, each root either lies in $U^{\perp }$ or, by subtracting a reflected vector of $U$, lies in $U$. These alternatives partition the roots into orthogonal sets. Irreducibility and spanning force $U=0$ or the whole space.

The relative position of Borel subgroups $B_1,B_2$ is the Weyl group index of their simultaneous conjugacy orbit. After moving the first subgroup to a fixed $B$, the second has the form $gB{g}^{-1}$ and its index is the double coset $BgB$. This identifies the diagonal action on pairs of flags with the Bruhat decomposition.

For the block-rotation maximal torus in the special orthogonal group $SO(2m+1)$, the Weyl group acts by permutations and independent inversions of $z_i$. Its invariant Laurent polynomials are the symmetric polynomials in $x_i=z_i+z_i^{-1}$, hence the polynomial ring in their elementary symmetric polynomials $e_1,\dots ,e_m$. Restriction of the group representation ring of a compact group is injective because every group element is conjugate into the torus and characters distinguish virtual representations. For the defining complexified representation $V$, the exterior-power characters satisfy ${\sum }_k{\chi }_{{\Lambda }^{k}V}{t}^k=(1+t){\prod }_i(1+x_{i}t+t^2)$. For $k\le m$, its coefficient is $e_k$ plus an integral linear combination of $e_j$ for $j<k$. Induction puts every $e_k$ in the restriction image, proving equality with the invariant ring and giving polynomial generators $[{\Lambda }^{1}V],\dots ,[{\Lambda }^{m}V]$.

The representation ring of finite-dimensional modules for a complex semisimple Lie algebra is the abelian group generated by module classes, with $[V\oplus U]=[V]+[U]$, and multiplication $[V][U]=[V\otimes U]$. Weyl complete reducibility theorem makes the classes of irreducible representations a free integral basis. The formal character map identifies this ring with the Weyl group invariant part of the group ring of a weight lattice. Injectivity follows from distinct highest weights; surjectivity follows by subtracting the character with each maximal dominant weight, whose highest-weight coefficient is one, from a Weyl-invariant finite sum.

// Target: semisimple-lie-algebra.bigb

The hyperplane perpendicular to a root of a root system is its root hyperplane, interpreted as the kernel of the associated linear functional on the Cartan space. The corresponding reflection is $s_{\alpha }(v)=v-2⟨v,\alpha ⟩\alpha /⟨\alpha ,\alpha ⟩$. For the B2 root system, the four distinct hyperplanes are the two coordinate axes and the two diagonals; their reflections generate the signed-permutation Weyl group of order eight.

Each root $\alpha$ of the SU(3) Lie algebra defines an SU(2) Lie algebra generated by $J_3=\alpha \cdot H$ and $J_{\pm }=\sqrt{2}{E}_{\pm \alpha }$ in the root-length-one convention. They obey $[J_3,J_{\pm }]=\pm J_{\pm }$ and $[J_{+},J_{-}]=2J_3$. These subalgebras explain root strings, Weyl group reflections and the integer Dynkin labels of highest weights.

// Target: lie-theory.bigb

Let $P_{a,b}$ be the convex hull of the Weyl group orbit of the highest weight $a{\omega }_1+b{\omega }_2$ for the SU(3) Lie algebra. Restrict the displayed formula to the weight lattice coset $p{\omega }_1+q{\omega }_2+$ root lattice. It describes successive inward shells: the outer boundary has weight multiplicity one, each inward shell increases it by one, and the final triangular region has the constant value $\min (p,q)+1$. When $p=q$ that region is a single point. A triangular diagram with $p=0$ or $q=0$ has multiplicity one throughout. This is a geometric form of the dominant weight multiplicity formula for sl3, transported by Weyl reflections.

// Target: lie-theory.bigb

In an irreducible root system with two root lengths, a short root has the smaller length with respect to a Weyl group invariant inner product.

For a Weyl group acting on a weight lattice, the displayed element of its group ring is alternating: applying $s\in W$ multiplies it by $\det (s)$, by reindexing the sum. It vanishes when $\nu$ is fixed by a root reflection. For the Weyl vector $\rho$, it is the Weyl denominator. The Weyl character formula is the ratio $A_{\lambda +\rho }/A_{\rho }$. These general root-system alternants specialize to monomial determinants in type A but are not restricted to that type.

The C3 root system consists of $\pm 2e_i$ and $\pm e_i\pm e_j$ in ${\mathbb{R}}^3$. Its Weyl group is the signed symmetric group on three coordinates, of order $48$. The reflecting planes are $x_i=0$ and $x_i=\pm x_j$. The open Weyl chamber $x_1>x_2>x_3>0$ has closure generated by $(1,0,0)$, $(1,1,0)$ and $(1,1,1)$. Sign choices and absolute-coordinate orderings give its $48$ translates. On the unit sphere the chamber is a spherical triangle with angles $\pi /2$, $\pi /3$, $\pi /4$, and hence area $\pi /12$; the areas of all $48$ triangles sum to $4\pi$.

The Weyl group of the Dn root system consists of coordinate permutations together with an even number of sign changes. Reflection in ${\epsilon }_i-{\epsilon }_j$ swaps coordinates $i,j$; reflection in ${\epsilon }_i+{\epsilon }_j$ swaps them and negates both. These generate all even signed permutations, giving order $2^{n-1}n!$.

Let $G$ be a connected compact Lie group, $T$ a maximal torus, and $W$ its Weyl group. With normalized Haar measures and $D(t)={\prod }_{\alpha >0}(1-e^{-\alpha }(t))$, every continuous class function satisfies ${\int }_{G}f(g)dg=|W{|}^{-1}{\int }_{T}f(t)|D(t){|}^{2}dt$. For a general continuous function, replace $f(t)$ by its average ${\int }_{G/T}f(gt{g}^{-1})d(gT)$. Conjugation from $G/T\times T$ covers the regular elements $|W|$ times, and its real Jacobian determinant is $|D(t){|}^2$.

For the unitary group, the diagonal maximal torus has coordinates $z_i=e^{i{\theta }_i}$ and Weyl group $S_n$. The Weyl integration formula weights independent uniform angles by $n{!}^{-1}{\prod }_{i<j}|z_i-z_j{|}^2$. The Vandermonde determinant $\det (z_i^{n-j})$ has squared torus integral $n!$, by Fourier orthogonality. Multiplying an irreducible character of highest weight $\lambda$ by this determinant gives an alternating Laurent polynomial. Its leading alternant has coefficient one; character orthogonality for compact groups forces every other alternant coefficient to vanish. This yields the quotient $\det (z_i^{{\lambda }_j+n-j})/\det (z_i^{n-j})$.

For an irreducible root system spanning $V$, the span $U$ of a nonzero root's Weyl group orbit is invariant, as is $U^{\perp }$. A root reflection preserving both subspaces must have its normal entirely in one of them: otherwise its rank-one correction mixes the subspaces. Thus a proper nonzero $U$ would partition all roots into mutually orthogonal subsets, contradicting irreducibility. Consequently some orbit root is nonorthogonal to any given root. Since Weyl reflections preserve lengths, the pairwise integrality restrictions for nonorthogonal roots also bound length ratios of orthogonal roots. All squared-length ratios lie in $\{1,2,1/2,3,1/3\}$; there cannot be three distinct lengths, since ratios two and three relative to a shortest root would yield the forbidden ratio $3/2$.

In a finite-dimensional sl2 Lie algebra representation, raising and lowering operators are nilpotent operators, so their exponentials are finite sums. In the defining two-dimensional representation, the displayed product is $\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$. It sends $h$ to $-h$ by conjugation, lifting the nontrivial Weyl group reflection. Its square acts on the irreducible highest-weight module $L(n)={\mathrm{Sym}}^n{\mathbb{C}}^2$ by $(-1{)}^n$. Thus a Weyl reflection can have a representative of order four on a representation.

// Target: linear-operator-theory.bigb