Higher Lie Characters in Algebra and Combinatorics

• Higher Lie characters are advanced algebraic constructs that generalize classical Lie characters through hybrid approaches, higher brackets, and polytope expansions.
• They play a crucial role in representation theory, algebraic combinatorics, and physics by enabling explicit tensor decompositions, asymptotic analysis, and efficient numerical integration.
• These characters bridge algebra, combinatorics, and geometry, supporting computations related to invariant polynomials, symmetric functions, and modular properties in various Lie theoretic contexts.

Higher Lie characters encompass a diverse family of character-theoretic constructions in representation theory, algebraic combinatorics, and Lie theory that generalize the classical concept of the Lie (or multilinear free Lie algebra) character. These characters arise from various generalizations of the Lie bracket or are induced from specific subgroups or modules associated with Lie groups, Lie algebras, or related combinatorial and geometric contexts. The theory of higher Lie characters connects deep algebraic structures, such as Weyl groups, invariant polynomials, and cocycles, with applications ranging from numerical analysis and quantum field theory to combinatorics of symmetric functions, tensor product decompositions, and the structure of modules over extended affine Lie algebras.

1. Classical and Hybrid Characters of Lie Groups

The classical character of a simple Lie group is defined via the Weyl character formula: $$\chi_\lambda(x) = \frac{S_{\lambda+\rho}(x)}{S_\rho(x)},$$ where $S_{\lambda+\rho}(x)$ is the Weyl–skew invariant orbit sum and $S_\rho(x)$ the Weyl denominator. In root systems with two root lengths, hybrid characters are constructed by introducing mixed sign homomorphisms $\sigma^s$, $\sigma^l$ on the Weyl group, modulating the signs according to whether a reflection is with respect to a short or long root. The hybrid $S$-functions are then

$$S^s_{\lambda+\rho^s}(x) = \sum_{w\in W} \sigma^s(w) e^{2\pi i\langle w(\lambda+\rho^s), x\rangle},$$

and similarly for $S^l$. The hybrid characters generalize ordinary characters: $$\chi^s_\lambda(x) = \frac{S^s_{\lambda+\rho^s}(x)}{S^s_{\rho^s}(x)},\qquad \chi^l_\lambda(x) = \frac{S^l_{\lambda+\rho^l}(x)}{S^l_{\rho^l}(x)}.$$ Hybrid characters generate the invariant polynomial ring and encode additional symmetry distinctions related to root lengths (Moody et al., 2012).

2. Higher Bracket Structures and Lie ℓ-ple Systems

Generalizations of the Lie bracket yield higher order algebraic structures, notably:

• Lie n-ple systems: An $n$-linear bracket $[X_1,\dots,X_n]$ that is skew-symmetric in its first $n-1$ entries and satisfies a cyclic antisymmetrization condition, generalizing Lie triple systems.
• Lie ℓ-ple systems: With $\ell = 2n-3$, the bracket $[X_1,\dots,X_{n-1},Y_1,\dots,Y_{n-2}]$ is required to be antisymmetric in each block and to satisfy a Filippov-type identity appropriate for multilinear operations.

A major construction uses the Kasymov trace form, a $2(n-1)$-linear invariant on a metric $n$-Leibniz algebra. For $n$-Leibniz algebra $\mathfrak{L}$,

$$k(X,Y) = \operatorname{Tr}(\operatorname{ad}_X\,\operatorname{ad}_Y)$$

for $(n-1)$-tuples $X,Y$. The $(2n-3)$-bracket is then defined via the process of "gluing" two adjoint maps using this form. These higher order brackets capture the algebraic structure of "higher" symmetries and appear in physical models generalizing the BLG theory and the Basu–Harvey equation (Azcarraga et al., 2013).

3. Combinatorial and Asymptotic Aspects in Symmetric Groups

In the symmetric group $S_n$, the classical Lie representation arises from the multilinear component of the free Lie algebra. Higher Lie characters generalize this construction: for $n$-cycles, they can be induced from linear characters of cyclic centralizers or via more intricate module constructions indexed by certain subsets of primes (Sundaram, 2021):

$$\mathrm{Lie}_n^S = \operatorname{ch} \left(e^{2\pi i Q_n / n}\uparrow_{C_n}^{S_n}\right)$$

where $Q_n$ selects maximal prime power divisors from a subset $S$ of primes.

The symmetric and exterior powers of these modules provide Thrall-type higher Lie modules whose Frobenius images are multiplicity-free sums of power sums. The associated generating functions,

$$H[\mathrm{Lie}^S](t) = \prod_{n\in P(S)} (1-t^n p_n)^{-1},$$

give rise to new families of Schur-positive symmetric functions and support plethystic relations between power sums, the conjugacy action, and the classical Lie representation (Sundaram, 2021).

Asymptotically, for random higher Lie characters or as $n\to\infty$, constituent multiplicities approach those of the regular character for Plancherel-typical Young diagrams. This is justified via the interplay of the Skiewicz–Weyman formula for multiplicities (via major index congruences), Swanson's error bounds, and Féray–Sniady's character ratio estimates, leading to the "regularization" of higher Lie characters in the asymptotic regime (Adin et al., 16 Sep 2025).

4. Polytope Expansion, Numerical Integration, and Cubature

Expanding Lie characters as sums over lattice polytopes offers a geometric interpretation of weight systems. Each character is written as

$\chi_\lambda = \sum_p A_{\lambda,p} B_p,$

with $B_p$ the generating function of the weights in a polytope and $A_{\lambda,p}$ their multiplicities (Walton, 2013). This expansion provides:

• Reduction of combinatorial complexity compared to the Weyl character/Kostant partition function formalism.
• Exact tensor product decompositions and explicit branching rules via optimal combinatorics of polytopes.
• Direct recovery of weight multiplicities from the polytope data.

In applied contexts, higher Lie characters manifest in Gaussian-type cubature formulas for integration of $W$-invariant functions. Cubature nodes are given by elements of finite order (EFO) in the torus, mapped via hybrid characters, leading to formulas of type: $$\int_{\Omega^s} f(X^s)\, (K^s)^{1/2}\, dX^s = \frac{1}{c_\mathfrak{g}\left(\frac{2\pi}{M + h^s}\right)^n} \sum_{X^s\in \mathcal{F}^s_{M+h^s}} f(X^s)\, \kappa^s(X^s)$$ where the right side achieves Gaussian (optimal) node count in the short root case (Moody et al., 2012).

5. Higher Lie Characters and Algebraic–Combinatorial Invariants

Higher Lie characters are central to the algebraic and combinatorial paper of permutation statistics and descent sets. For a conjugacy class $C_\lambda$ of a given cycle type $\lambda$, the induced higher Lie character $\chi^\lambda$ encodes descent statistics, and hook multiplicity expansions

$M(x) = \sum_{k=0}^{n-1} m_k x^k$

(where $m_k$ is the multiplicity in the hook partition $(n-k,1^k)$) govern the existence of cyclic descent extensions. The divisibility condition $1+x\mid M(x)$ and positivity of $M(x)/(1+x)$ are necessary and sufficient for such an extension, with certain square-free cycle types being obstructions (Adin et al., 2019).

These connections provide a bridge between the representation theory of $S_n$ and the enumerative or probabilistic structures embedded in symmetric function theory, especially via quasisymmetric functions and Schur positivity.

6. Affine, Super, and Extended Lie Algebras

Higher Lie characters and their analogues extend to characters of modules over affine Lie algebras (including parafermionic coset theories, with character formulas expressing level two string functions as generalized Rogers–Ramanujan (GRR) $q$-series) and over extended affine Lie algebras. These characters are often quotients of $q$-series by powers of the Dedekind eta function and encode modular properties of relaxed highest weight modules, with deep links to BRST reductions and minimal nilpotent orbits (Genish et al., 2014, Gepner, 2014, Kawasetsu, 2020).

In the context of extended affine Lie algebras, diagonal Cartan automorphisms yield core-characters that satisfy the multiplicativity conditions associated with the Lie theoretic structure. The behavior of such characters may differ significantly from the affine/finite-dimensional cases, especially in the subtlety of their extension to root lattices (Azam, 7 Jun 2024).

7. Applications, Structure, and Future Directions

• Numerical Analysis: Gaussian cubature and efficient function approximation on symmetric domains via hybrid/higher Lie characters.
• Algebraic Combinatorics: Asymptotic behavior and decomposition of higher Lie characters, connections with major index statistics, cyclic extensions, and Schur positivity of symmetric functions.
• Physics: Representation-theoretic interpretation of gauge models with higher brackets, especially in relation to M2-brane models and coset conformal field theories.
• Operator Algebras and Dynamics: Character rigidity, property (T), structure of C*-algebras attached to lattices in higher-rank Lie groups, and invariant theory for stationary characters (Boutonnet et al., 2019).

Open questions include explicit decompositions of higher Lie characters in irreducible bases in non-asymptotic regimes (Thrall's problem), further characterization of “proper” characters for root enumerator functions in classical Weyl groups (Adin et al., 2023), the extension theory of characters in higher nullity Lie algebras, and deeper plethystic and combinatorial identities among associated symmetric functions.

Context / Structure	Construction/Formula	Significance
Classical Lie character	Weyl formula: $\chi_\lambda$	Traditional representation theory
Hybrid / higher characters	Mixed sign, induced character, or higher bracket	Distinguish root lengths, encode symmetries
Symmetric group ($S_n$)	$\mathrm{Lie}_n^S$, plethysm, major index statistics	Combinatorics, Schur and power sum expansions
Polytope expansion	$\chi_\lambda = \sum_p A_{\lambda,p} B_p$	Geometric–combinatorial simplification
Affine/coset/parafermionic	$q$-series, string functions, GRR identities	CFT, modular invariance, combinatorial models
Extended affine Lie algebras	Diagonal Cartan automorphism yields core-character	Automorphism/structure theory, higher nullity

In sum, higher Lie characters formalize and generalize the structure of symmetries and representations across a wide spectrum of mathematical and physical theories, providing tools for explicit decomposition, asymptotic analysis, and efficient computation in contexts marked by high symmetry and combinatorial richness.