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An automorphism of a root system $\Phi \subset E$ is an orthogonal transformation of $E$ mapping $\Phi$ onto itself. The Weyl group is a subgroup of this automorphism group; additional automorphisms can act nontrivially on a Dynkin diagram while preserving a root basis.

A BN-pair in a group $G$ consists of subgroups $B,N$ such that $G=⟨B,N⟩$, $H=B\cap N$ is normal in $N$, and $W=N/H$ has distinguished involutory generators satisfying the Bruhat multiplication axioms. The quotient $W$ is its Weyl group.

For a connected reductive algebraic group, a Borel subgroup $B$, and a maximal torus $T\subseteq B$, the Weyl group $W=N_G(T)/T$ indexes the double cosets:

The Weyl group of the Cn root system is the signed symmetric group. A Weyl reflection in $2{\epsilon }_i$ changes the sign of coordinate $i$; one in ${\epsilon }_i-{\epsilon }_j$ interchanges coordinates $i,j$; one in ${\epsilon }_i+{\epsilon }_j$ interchanges and negates both coordinates. These generate all independent sign changes and all permutations, giving order $2^{n}n!$.

The scalar potential is a positive sum of squared commutators of the six adjoint scalar fields. It vanishes precisely when they commute. For a compact gauge group, simultaneous conjugation puts a commuting tuple in a Cartan subalgebra. Its arbitrary $6r$ real coordinates are classical flat directions of a scalar potential, with residual identification by the Weyl group. For $SU(N)$ the simultaneous eigenvalue vectors also obey the tracelessness constraint. Generic values break the gauge group to its maximal torus.

For a connected reductive algebraic group and a Weyl group element $w$, this locally closed subset of the flag variety consists of flags in relative position $w$ to their Frobenius image. For rational $B$ it is ${\mathcal{L}}_F^{-1}(B\dot{w}B)/B$, using the twisted right action identity of the Lang map. Its dimension is the length of $w$.

For a finite-dimensional irreducible highest-weight representation with highest weight $\lambda$, every weight in the Weyl group orbit $W\lambda$ is extremal and has weight multiplicity one.

A finite group of Lie type is obtained from rational points of a connected reductive algebraic group with a suitable Frobenius endomorphism of an algebraic group, often followed by central quotient or passage to a derived subgroup. Its complex representations combine finite-group character theory with the geometry of Borel subgroups, maximal algebraic tori, and the Weyl group.

The formal character of a weight module for a minuscule representation of highest weight $\lambda$ is $\mathrm{ch}V(\lambda )={\sum }_{\mu \in W\lambda }{e}^{\mu }$. The sum is over distinct points of the Weyl group orbit, so $\dim V(\lambda )=|W\lambda |$.

The group ring of a weight lattice $P$ consists of finite sums ${\sum }_{\lambda }{a}_{\lambda }e(\lambda )$ with integer coefficients, where $e(\lambda )e(\mu )=e(\lambda +\mu )$. A finite-dimensional Lie algebra representation has formal character ${\sum }_{\lambda }(\dim V_{\lambda })e(\lambda )$ in this ring. The Weyl group acts by $e(\lambda )\mapsto e(w\lambda )$. The ambient ring need not consist of genuine characters; the representation ring of a semisimple Lie algebra of finite-dimensional modules for a complex semisimple Lie algebra identifies with the invariant subring $\mathbb{Z}[P{]}^W$.

// Target: semisimple-lie-algebra.bigb

For a chosen positive system of a root system $\Pi$, the inversion set isFor a finite Weyl group, $|N(w)|=\ell (w)$.

Its cardinality is the Coxeter length; this root-sign definition should not be silently identified with an unspecified permutation inversion convention.

Multiply the Weyl character formula by the generating series of the Kostant partition function. In each numerator summand, the coefficient of $e^{\mu }$ occurs when $\nu =w(\lambda +\rho )-(\mu +\rho )$. Summing those coefficients with the Weyl group signs proves the displayed weight multiplicity formula. Although $\rho$ need not itself be an integral group weight, $w\rho -\rho$ is in the root lattice, so each partition-function argument is an integral weight.

The longest element of a finite Weyl group is the element sending all positive roots of a root system to negative roots. It is the longest element of a finite Coxeter group. For the G2 root system it is $-I$: its twelve roots form two hexagons, and the dihedral reflection group of a root system contains the rotation through $\pi$. It determines the highest weight of a dual representation.

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

A finite-dimensional irreducible representation of a complex semisimple Lie algebra is minuscule if its weights form one Weyl group orbit. Every weight multiplicity is then one, because multiplicities are invariant under the Weyl group and the highest weight has multiplicity one.

By dominant root-lattice highest weights have zero weight, a minuscule weight in the root lattice has zero in its Weyl group orbit. Since every element of the Weyl group is invertible, the highest weight must itself be zero. Thus equality of the weight lattice and root lattice rules out every nontrivial minuscule representation without requiring a classification of root systems.

A maximal torus in a compact connected Lie group is a compact connected abelian Lie subgroup maximal among such subgroups. By the connected abelian classification it is isomorphic to a product of circles. Such subgroups exist: choose one of largest dimension among compact connected abelian subgroups; a proper inclusion of connected Lie subgroups would increase dimension. Write $\mathfrak{t}$ for its Lie algebra.

Here are the main steps of the conjugacy assertion, with the critical arguments supplied. Average an inner product on $\mathfrak{g}$ over the adjoint action to obtain an $\mathrm{Ad}(G)$-invariant positive definite inner product. It defines a bi-invariant Riemannian metric. Compactness makes this metric complete, and the Hopf-Rinow theorem joins the identity to any $g$ by a geodesic. For a bi-invariant metric the Levi-Civita connection on left-invariant fields is ${\nabla }_{X}Y=[X,Y]/2$, so the geodesic starting with $X$ is $\exp (tX)$. Consequently every $g$ has the form $\exp X$.

The Lie-algebra centralizer of $\mathfrak{t}$ is exactly $\mathfrak{t}$. If $X$ commutes with $\mathfrak{t}$, its one-parameter subgroup commutes with $T$. The closure of the subgroup generated by $T$ and $\exp (\mathbb{R}X)$ is compact, connected and abelian, so it is a torus; maximality forces it to equal $T$, giving $X\in \mathfrak{t}$. The torus action on ${\mathfrak{g}}_{\mathbb{C}}$ decomposes into finitely many character weight spaces. The zero-weight space is ${\mathfrak{t}}_{\mathbb{C}}$. Choose $H\in \mathfrak{t}$ outside the finitely many kernels of the nonzero weight differentials. Then the centralizer of $H$ is exactly $\mathfrak{t}$.

For an arbitrary $X\in \mathfrak{g}$, the continuous function $g\mapsto ⟨{\mathrm{Ad}}_{g}X,H⟩$ attains a maximum. Put $Y={\mathrm{Ad}}_{g_0}X$ at a maximizer. Differentiating along $\exp (sZ)g_0$ givesHence $[Y,H]=0$, so $Y\in \mathfrak{t}$. Exponentiating gives $g_0(\exp X)g_0^{-1}=\exp Y\in T$. This adjoint-orbit criterion for maximal-torus conjugacy proves

Restriction of representations is a ring homomorphism $\mathrm{Res}:R(G)\to R(T)$. If a virtual representation restricts to zero, its character vanishes on $T$. Since characters are class functions and every element is conjugate into $T$, it vanishes on all of $G$. The injectivity of the character map proved in Question 3 then givesThe Weyl group is $W=N_G(T)/T$. For $n\in N_G(T)$, the operator $\rho (n)$ intertwines the restriction of $\rho$ with its twist by $t\mapsto nt{n}^{-1}$. Therefore every restricted representation, and every virtual difference of them, is fixed by $W$:

For the special orthogonal group $G=SO(2m+1)$, takewhere $R_{\theta }$ is a real planar rotation. It is a maximal torus: its Lie-algebra centralizer consists of the independent infinitesimal rotations in those planes. Its character lattice is ${\mathbb{Z}}^m$, so, writing $z_i=e^{i{\theta }_i}$,Indeed commuting unitary torus matrices diagonalize simultaneously into one-dimensional characters, and the characters of the circle are exactly the integral powers.

The normalizer permutes the real rotation planes and can reverse each angle independently. Permutations of whole two-dimensional blocks have determinant one. A reflection in one rotation plane reverses its angle and has determinant minus one; combining it with reflection of the remaining fixed line gives an element of $SO(2m+1)$. Thus all independent inversions occur. Conversely a normalizer must permute the distinct nontrivial torus character pairs $\{\pm e_i\}$ and preserve the fixed line, so there are no additional actions. The kernel of this action is exactly $T$, giving

It follows that the restriction identifies $R(SO(2m+1))$ with a subring of the signed-permutation-invariant Laurent polynomials. In this case we can prove equality. Invariance under individual inversions first givesFor one variable, every invariant is an integral combination of $1$ and $z^k+z^{-k}$, and these satisfy the recurrence $s_{k+1}=x{s}_k-s_{k-1}$. Applying this one coordinate at a time proves the assertion over the other Laurent coefficient rings. Permutation invariance and the Fundamental theorem of symmetric polynomials then givewhere $e_k$ are the elementary symmetric polynomials.

Let $V$ be the complexification of the defining $(2m+1)$-dimensional real representation. Its torus weights are $1,z_i,z_i^{-1}$. The characters of its exterior powers obeyFor $k\le m$, choosing $j$ linear terms and $r$ quadratic terms shows that the coefficient $c_k={\chi }_{{\Lambda }^{k}V}$ isIts leading term is $e_k$ with coefficient one; all other terms use $e_j$ with $j<k$. By induction $e_k$ lies in the restriction image, because $c_k$ is the character of a genuine representation and all smaller $e_j$ already lie in the image. Hence every invariant polynomial lies there, proving the representation ring of an odd-dimensional special orthogonal group descriptionThe triangular change of generators is integral, so $[{\Lambda }^{1}V],\dots ,[{\Lambda }^{m}V]$ are algebraically independent polynomial generators. For $m=1$, $c_1=1+z+z^{-1}$ and this reduces to $R(SO(3))=\mathbb{Z}[V_3]$, with the three-dimensional defining representation as generator. These are genuine representations of the orthogonal group, not the half-integral spin weights of its simply connected double cover.

Work over $\mathbb{C}$ with a finite-dimensional semisimple Lie algebra $\mathfrak{g}$. Choose a Cartan subalgebra $\mathfrak{h}$, its root system $R\subset {\mathfrak{h}}^{\ast }$, a positive system $R^{+}$ and its simple roots. Let $W$ be the Weyl group, generated by the corresponding root reflections, and put $\rho =\frac{1}{2}{\sum }_{\alpha \in R^{+}}\alpha$, the Weyl vector. The finite-dimensional irreducible highest-weight representation $L(\lambda )$ is indexed by a dominant integral weight $\lambda$.

If $P$ is the weight lattice, its group ring has formal symbols $e^{\mu }$ with multiplication $e^{\mu }{e}^{\nu }=e^{\mu +\nu }$. The formal character iswhere $L(\lambda {)}_{\mu }=\{v:Hv=\mu (H)v\text{ for all }H\in \mathfrak{h}\}$ is a weight space. For a weight $\nu$, define the Weyl alternant $A_{\nu }={\sum }_{w\in W}\det (w)e^{w\nu }$; the sign $\det (w)=(-1{)}^{\ell (w)}$ is the determinant of $w$ on the real root span, equivalently the parity of its expression in simple reflections.

The Weyl character formula isThe second denominator is the Weyl denominator formula. The quotient can initially be taken in the fraction field of the group ring; the formula says it is the finite character on the left, so its apparent singularities are removable. The shifts by $\rho$ make $\lambda +\rho$ strictly dominant.