The Brauer–Klimyk and Racah–Speiser formulas #
Let L be a finite-dimensional Lie algebra with non-degenerate Killing form over an algebraically
closed field of characteristic zero, H a Cartan subalgebra, and b a base of its root system.
Weyl-alternating inversion reads the isotypic multiplicities of a finite-dimensional L-module from
its formal character times the Weyl denominator Δ. Applied to the tensor product L(λ) ⊗ M of a
finite-dimensional irreducible L(λ) with an arbitrary finite-dimensional L-module M, it gives
a closed form for the multiplicity [L(λ) ⊗ M : L(ν)] of L(ν) in L(λ) ⊗ M. Formal characters
are multiplicative on tensor products (TauCeti.formalCharacter_tensor), and the Weyl character
formula rewrites the first factor as ch L(λ) · Δ = N(λ)
(TauCeti.formalCharacter_mul_weylDenominator_eq_weylNumerator), so
[L(λ) ⊗ M : L(ν)] = [e^ν] (ch M · N(λ)) = ∑_{w ∈ W} sgn(w) · [e^{ν - w ⬝ λ}] ch M.
This is the Brauer–Klimyk formula: to compute the multiplicity of L(ν) in L(λ) ⊗ M, shift
the weights of M by λ, fold those that land outside the dominant chamber back into it by the dot
action of the Weyl group with signs sgn(w), and read off the resulting coefficient at ν. Its
case M = L(μ) is the Racah–Speiser formula for the tensor multiplicity
c^ν_{λμ} = TauCeti.tensorMultiplicity b λ μ ν.
Main results #
TauCeti.isotypicMultiplicity_tensor_eq_coeff_formalCharacter_mul_weylNumeratorandTauCeti.isotypicMultiplicity_tensor_eq_sum_weylSign_mul_formalCharacter_coeff: the Brauer–Klimyk formula forL(λ) ⊗ M, in group-algebra form and expanded over the Weyl group.TauCeti.tensorMultiplicity_eq_coeff_formalCharacter_mul_weylNumeratorandTauCeti.tensorMultiplicity_eq_sum_weylSign_mul_formalCharacter_coeff: the Racah–Speiser formula, the caseM = L(μ).
References #
- N. Bourbaki, Groupes et algèbres de Lie, Ch. VIII, §9, Exercise 2 (the Brauer–Klimyk formula and its specialization to minuscule weights).
- J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, GTM 9, Ch. VI, §24, Exercise 9.
The Brauer–Klimyk formula, in group-algebra form: for dominant integral weights lam and
nu and a finite-dimensional L-module M, the multiplicity of L(nu) in L(lam) ⊗ M is the
coefficient at nu of ch M · N(lam).
This combines Weyl-alternating inversion
(TauCeti.coeff_formalCharacter_mul_weylDenominator_eq_isotypicMultiplicity) on L(lam) ⊗ M,
multiplicativity of formal characters (TauCeti.formalCharacter_tensor), and the Weyl character
formula ch L(lam) · Δ = N(lam).
The Brauer–Klimyk formula, expanded over the Weyl group: for dominant integral weights
lam and nu and a finite-dimensional L-module M, the multiplicity of L(nu) in
L(lam) ⊗ M is ∑_{w ∈ W} sgn(w) · [e^{nu - w ⬝ lam}] ch M.
The Racah–Speiser formula, in group-algebra form: for dominant integral weights lam,
mu, and nu, the tensor multiplicity c^nu_{lam mu} of L(nu) in L(lam) ⊗ L(mu) is the
coefficient at nu of ch L(mu) · N(lam).
This is the Brauer–Klimyk formula
TauCeti.isotypicMultiplicity_tensor_eq_coeff_formalCharacter_mul_weylNumerator for
M = L(mu).
The Racah–Speiser formula, expanded over the Weyl group: for dominant integral weights
lam, mu, and nu, c^nu_{lam mu} = ∑_{w ∈ W} sgn(w) · [e^{nu - w ⬝ lam}] ch L(mu).
This is the Brauer–Klimyk formula
TauCeti.isotypicMultiplicity_tensor_eq_sum_weylSign_mul_formalCharacter_coeff for
M = L(mu).