Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 2 / 3

The ambient group ring of a weight lattice isAddition is coefficientwise, and multiplication uses addition in the weight lattice. For a finite-dimensional Lie algebra representation $V={⨁}_{\lambda }{V}_{\lambda }$, its formal character isThe formal character records the dimensions of all weight spaces. The ambient lattice ring should be distinguished from the representation ring of a semisimple Lie algebra, which identifies with its Weyl-invariant subring.

For the ${\mathfrak{sl}}_3$ module ${\Gamma }_{1,2}$, $L_1+L_2+L_3=0$ and the highest weight is $L_1-2L_3={\omega }_1+2{\omega }_2$. The decomposition $V\otimes {\mathrm{Sym}}^2{V}^{\ast }={\Gamma }_{1,2}\oplus V^{\ast }$ givesCollecting equal weights yieldsThe six first weights and three second weights have multiplicity one; each of the three weights $-L_i$ has multiplicity two. Their total is $6+3+6=15$, as required.

For $v\in V_{\lambda }$ and $w\in W_{\mu }$, the tensor product of Lie algebra representations acts byThus its weight space of weight $\nu$ isSubstituting this into the formal character and using $e(\lambda +\mu )=e(\lambda )e(\mu )$ provesEvery sum is finite for the Lie algebra representations under consideration.

The Weyl group acts on the weight lattice by its root reflections and extends to ring automorphisms of the group ring of a weight lattice:Weights and their multiplicities in a finite-dimensional Lie algebra representation of a semisimple Lie algebra are permuted by this action, so its formal character is Weyl-invariant.

Choose a set ${\Phi }^{+}$ of positive roots and putHere ${\omega }_i$ are the fundamental weights, and $\det (w)=(-1{)}^{\ell (w)}$, where the Coxeter length $\ell (w)$ counts positive roots carried to negative ones. For a dominant integral weight $\lambda$, let $L(\lambda )$ denote the finite-dimensional Irreducible Lie algebra representation of highest weight $\lambda$. The Weyl character formula isThe second denominator expression is the Weyl denominator formula. Both alternants change sign under each root reflection; their quotient is invariant. Although written as a fraction, the theorem asserts that it is a finite element of $\mathbb{Z}[{\Lambda }_W]$ with the nonnegative coefficients given by weight multiplicities. The highest-weight classification specifies which $\lambda$ label the irreducibles, while the root system and choice of positive roots determine every other term.

Write $u={\sum }_{\lambda }{a}_{\lambda }e(\lambda )$ with finite support. The root reflection is an involution, and $s_{\alpha }u=-u$ implies $a_{s_{\alpha }\lambda }=-a_{\lambda }$. Any fixed weight has coefficient zero. For each nonfixed orbit choose its representative $\lambda$ withIntegrality of $m$ follows because $\lambda$ belongs to the weight lattice; positivity is ensured by our choice of orbit representative. Pairing the orbit coefficients givesFor each pair, elementary finite telescoping showsThere are finitely many chosen orbits, and each inner sum is finite. This proves the root-binomial divisibility of reflection anti-invariants:

For precision, the geometric-series expression is interpreted coefficientwise. Its coefficient at a weight $\nu$ is ${\sum }_{n\ge 0}{a}_{\nu -n\alpha }$, which is a finite sum because $u$ has finite support. On each coset of $\mathbb{Z}\alpha$, reflection pairs the coefficients with opposite signs, so their total is zero. These cumulative sums vanish both before the smallest occupied weight and after the largest occupied weight. Only finitely many cosets are occupied, hence the product of the formal series with $u$ has finite support and equals the boxed quotient.

The individual terms $e(n\alpha )u$ need not vanish for large $n$. For example, $u=e(\alpha )-e(-\alpha )$ is anti-invariant, and its quotient is $-1-e(-\alpha )$. Thus “finite sum” means a finite result after coefficientwise cancellation, not an actually truncated geometric series. No analytic convergence assumption is involved.