Nested Weyl Character Formulas

• Nested Weyl-type character formulas are identities that combine classical Weyl-group alternation with inner layers like BGG resolutions, Demazure operators, and combinatorial sums.
• They are applied across quantum affine algebras, Borcherds superalgebras, and Lie superalgebras, offering resolution-theoretic, operator-theoretic, and correction-theoretic formulations.
• These formulas reconstruct characters through an outer Weyl alternation paired with inner finite sums or deformations, enabling applications from lattice polytope decompositions to Tokuyama-type analyses.

Searching arXiv for the specified paper and closely related work on Weyl-type and nested character formulas. Nested Weyl-type character formulas are character identities in which the classical Weyl-group alternation is combined with additional internal layers such as BGG-type resolutions, Demazure-operator compositions, imaginary-root correction sums, polyhedral vertex expansions, or combinatorial pattern sums. In the recent and adjacent literature, this label does not designate a single formalism; rather, it describes a recurring structural phenomenon across quantum affine algebras, Borcherds-type algebras, Lie superalgebras, weight polytopes, and Tokuyama-type deformations, where a character is assembled by an outer Weyl-symmetrization together with inner finite sums, iterated operators, or inductive combinatorial strata (Brito et al., 2017, Fan et al., 2024, Walton, 2021, Makhlin, 2014, Chmutov et al., 2014, Brubaker et al., 2014, Friedberg et al., 2014, Pal et al., 12 May 2025).

1. Structural meaning of “nested” in Weyl-type formulas

A first sense of nesting is resolution-theoretic. In this form, the irreducible character is obtained as an alternating sum of characters of better-understood modules, so that the Weyl-type formula is produced by a homological tower rather than by a single denominator identity. A second sense is operator-theoretic: one replaces the global Weyl numerator-denominator mechanism by a composition of simple-root or positive-root operators, each of which adds one string, one boundary strip, or one branching layer. A third sense is correction-theoretic: the usual Weyl sum over real roots is retained, but it is multiplied by a finite auxiliary sum indexed by imaginary roots, holes, or atypical blocks. A fourth sense is combinatorial: one rewrites a character or deformed character as a sum over Gelfand-Tsetlin patterns, lattice-model states, or tangent cones, and the nesting is realized by induction on rank, row, or boundary path (Fan et al., 2024, Walton, 2021, Makhlin, 2014).

These constructions are related by a common architecture. The outermost layer is typically an alternating Weyl-group action, such as $\sum_{w\in W}\mathrm{sgn}(w)\,w(\cdots)$ or $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$. Inside that outer layer, one finds one of several inner mechanisms: a local Weyl-module resolution, a correction factor $S_\lambda$ over mutually orthogonal imaginary roots, a nested product of Demazure operators, a Brion sum over tangent cones, or a combinatorial sum over patterns or six-vertex states. The literature therefore uses “nested Weyl-type” to indicate layered organization rather than a unique universal formula. This also explains why the same terminology appears in settings with very different representation-theoretic input.

2. Resolution-theoretic formulas for quantum affine $\mathfrak{sl}_{n+1}$

For quantum affine $\mathfrak{sl}_{n+1}$, one important instance arises in the family of prime irreducible representations studied in connection with the work of D. Hernandez and B. Leclerc. These representations are described by highest weights that are products of distinct fundamental weights, with parameters chosen so that the representation is minimal by parts. The key result is that such representations admit a BGG-type resolution in which the Verma module is replaced by the local Weyl module. This leads to a closed Weyl character formula for the irreducible representation as an alternating sum of characters of local Weyl modules (Brito et al., 2017).

In this setting, the nesting is carried by the resolution itself. The irreducible module is not described directly by a single quotient formula; instead, its character is reconstructed through successive local Weyl-module terms with alternating signs. The same result has two further reformulations. In the language of cluster algebras, the Weyl character formula describes an arbitrary cluster variable in terms of the generators $x_1,\cdots,x_n,x_1',\cdots,x_n'$ of an appropriate cluster algebra. In addition, the character of a prime level two Demazure module is exhibited as an alternating linear combination of level one Demazure modules (Brito et al., 2017).

This resolution-theoretic form is significant because it relocates the Weyl-type phenomenon from the classical Verma-module setting to the local Weyl-module setting. A plausible implication is that the term “nested” here refers less to a nested summation index than to a homological filtration whose Euler characteristic is the irreducible character.

3. Imaginary-root corrections in Borcherds and Borcherds-Bozec settings

For Borcherds-Bozec superalgebras, the Weyl-Kac-type character formula for irreducible highest-weight modules with dominant integral highest weights has the form

$$(e^\rho R)\,\mathrm{ch}\,V(\lambda) = \sum_{w\in W}\mathrm{sgn}(w)\;w\bigl(e^{\lambda+\rho}S_\lambda\bigr),$$

or equivalently

$$\mathrm{ch}\,V(\lambda) = e^{-\rho}R^{-1}\sum_{w\in W}\mathrm{sgn}(w)\;w\bigl(e^{\lambda+\rho}S_\lambda\bigr).$$

Here $R$ is the super-denominator, and $S_\lambda$ is a finite correction factor built from imaginary simple roots. More precisely, $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$0, where $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$1 is assembled from non-isotropic imaginary simple roots in $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$2 and isotropic imaginary simple roots in $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$3, subject to orthogonality and annihilation conditions relative to $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$4. The coefficients $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$5 split into $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$6 and $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$7, with the non-isotropic part governed by the coefficients $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$8 from $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$9, and the isotropic part involving recursively defined integers $S_\lambda$0 (Fan et al., 2024).

The paper explicitly interprets this as a layered formula. The outermost layer is the classical Weyl-symmetrization over $S_\lambda$1 acting on $S_\lambda$2. Inside, one has the finite sum $S_\lambda$3, itself split into the non-isotropic imaginary subsystem $S_\lambda$4 and the isotropic imaginary subsystem $S_\lambda$5. The coefficients from $S_\lambda$6 are described as “bosonic” layers arising from $S_\lambda$7 expansions, while those from $S_\lambda$8 are described as “fermionic” layers coming from $S_\lambda$9 factors refined by the recursive $\mathfrak{sl}_{n+1}$0. The formula is therefore nested both algebraically and combinatorially (Fan et al., 2024).

A related Borcherds-Kac-Moody framework generalizes Weyl-Kac-Borcherds-type formulas to arbitrary highest-weight modules and introduces the cone $\mathfrak{sl}_{n+1}$1 and the notion of “holes.” In the summary provided, the master formula for any integrable simple $\mathfrak{sl}_{n+1}$2 with $\mathfrak{sl}_{n+1}$3 is presented as

$\mathfrak{sl}_{n+1}$4

with

$\mathfrak{sl}_{n+1}$5

The same work defines higher-order Verma modules

$\mathfrak{sl}_{n+1}$6

and, in rank two, reduces the character to a finite numerator over the universal denominator

$\mathfrak{sl}_{n+1}$7

where the numerator contains four, six, or fewer terms with coefficients $\mathfrak{sl}_{n+1}$8 or $\mathfrak{sl}_{n+1}$9, indexed by solutions of a Diophantine equation or, equivalently, by the Casimir-norm equality condition

$\mathfrak{sl}_{n+1}$0

This places the nested Weyl-type structure in direct relation with independent subsets of imaginary nodes, maximal vectors, and finitely many Casimir-compatible weights (Pal et al., 12 May 2025).

4. Demazure operators, weight polytopes, and Brion towers

A second major realization of nesting replaces the usual Weyl formula by iterated Demazure operators. For a simple Lie algebra with simple roots $\mathfrak{sl}_{n+1}$1, the primitive Demazure operator is

$\mathfrak{sl}_{n+1}$2

and for a reduced decomposition $\mathfrak{sl}_{n+1}$3 one sets $\mathfrak{sl}_{n+1}$4. The general Demazure character formula is

$\mathfrak{sl}_{n+1}$5

with $\mathfrak{sl}_{n+1}$6 the longest Weyl-group element. Equivalently, if $\mathfrak{sl}_{n+1}$7, then

$\mathfrak{sl}_{n+1}$8

In this framework, each successive application of a simple-root operator grows a string of weights parallel to one root direction. The nested composition $\mathfrak{sl}_{n+1}$9 is therefore a string-by-string analogue of the Weyl denominator expansion (Walton, 2021).

Walton’s formulas for lattice sums of weight polytopes make this nesting explicit for rank-$x_1,\cdots,x_n,x_1',\cdots,x_n'$0 types $x_1,\cdots,x_n,x_1',\cdots,x_n'$1, $x_1,\cdots,x_n,x_1',\cdots,x_n'$2, $x_1,\cdots,x_n,x_1',\cdots,x_n'$3, and for $x_1,\cdots,x_n,x_1',\cdots,x_n'$4. If the positive roots along a chosen boundary path are ordered as $x_1,\cdots,x_n,x_1',\cdots,x_n'$5, then the lattice-polytope sum $x_1,\cdots,x_n,x_1',\cdots,x_n'$6 can be written as $x_1,\cdots,x_n,x_1',\cdots,x_n'$7, where $x_1,\cdots,x_n,x_1',\cdots,x_n'$8 is a nested operator built from factors of the form $x_1,\cdots,x_n,x_1',\cdots,x_n'$9 and interspersed reflections. For $$(e^\rho R)\,\mathrm{ch}\,V(\lambda) = \sum_{w\in W}\mathrm{sgn}(w)\;w\bigl(e^{\lambda+\rho}S_\lambda\bigr),$$0,

$$(e^\rho R)\,\mathrm{ch}\,V(\lambda) = \sum_{w\in W}\mathrm{sgn}(w)\;w\bigl(e^{\lambda+\rho}S_\lambda\bigr),$$1

The paper states that the rank-$$(e^\rho R)\,\mathrm{ch}\,V(\lambda) = \sum_{w\in W}\mathrm{sgn}(w)\;w\bigl(e^{\lambda+\rho}S_\lambda\bigr),$$2 and $$(e^\rho R)\,\mathrm{ch}\,V(\lambda) = \sum_{w\in W}\mathrm{sgn}(w)\;w\bigl(e^{\lambda+\rho}S_\lambda\bigr),$$3 cases strongly suggest a universal pattern and concludes that “Demazure-type formulas can be written,” conjecturing a universal nested-Demazure formula for arbitrary simple Lie algebras (Walton, 2021).

A polyhedral version of the same phenomenon appears through Brion’s theorem applied to the Gelfand-Tsetlin chain

$$(e^\rho R)\,\mathrm{ch}\,V(\lambda) = \sum_{w\in W}\mathrm{sgn}(w)\;w\bigl(e^{\lambda+\rho}S_\lambda\bigr),$$4

For the $$(e^\rho R)\,\mathrm{ch}\,V(\lambda) = \sum_{w\in W}\mathrm{sgn}(w)\;w\bigl(e^{\lambda+\rho}S_\lambda\bigr),$$5th Gelfand-Tsetlin polytope $$(e^\rho R)\,\mathrm{ch}\,V(\lambda) = \sum_{w\in W}\mathrm{sgn}(w)\;w\bigl(e^{\lambda+\rho}S_\lambda\bigr),$$6, one obtains

$$(e^\rho R)\,\mathrm{ch}\,V(\lambda) = \sum_{w\in W}\mathrm{sgn}(w)\;w\bigl(e^{\lambda+\rho}S_\lambda\bigr),$$7

At the top stage, only $$(e^\rho R)\,\mathrm{ch}\,V(\lambda) = \sum_{w\in W}\mathrm{sgn}(w)\;w\bigl(e^{\lambda+\rho}S_\lambda\bigr),$$8 simplicial vertices survive after specialization in the regular-$$(e^\rho R)\,\mathrm{ch}\,V(\lambda) = \sum_{w\in W}\mathrm{sgn}(w)\;w\bigl(e^{\lambda+\rho}S_\lambda\bigr),$$9 case, and their tangent-cone transforms are exactly the Weyl summands. Iterating the stage-by-stage Brion decomposition yields a fully nested sum over chains

$$\mathrm{ch}\,V(\lambda) = e^{-\rho}R^{-1}\sum_{w\in W}\mathrm{sgn}(w)\;w\bigl(e^{\lambda+\rho}S_\lambda\bigr).$$0

whose expansion collapses to the usual Weyl character formula for $$\mathrm{ch}\,V(\lambda) = e^{-\rho}R^{-1}\sum_{w\in W}\mathrm{sgn}(w)\;w\bigl(e^{\lambda+\rho}S_\lambda\bigr).$$1 (Makhlin, 2014).

Taken together, the Demazure and Brion approaches show that a Weyl character can be reconstructed by repeatedly adding local data: strings of weights in the operator picture, or tangent-cone contributions in the polyhedral picture. This suggests a strong relation between nested operator formulas and nested polytope decompositions.

5. Lie-superalgebra formulas and blockwise Weyl summation

For $$\mathrm{ch}\,V(\lambda) = e^{-\rho}R^{-1}\sum_{w\in W}\mathrm{sgn}(w)\;w\bigl(e^{\lambda+\rho}S_\lambda\bigr).$$2, a Weyl-type character formula is available for the class of piecewise disconnected modules, or PDC modules. The setup uses the standard Cartan, the Weyl group $$\mathrm{ch}\,V(\lambda) = e^{-\rho}R^{-1}\sum_{w\in W}\mathrm{sgn}(w)\;w\bigl(e^{\lambda+\rho}S_\lambda\bigr).$$3, the Weyl vector

$$\mathrm{ch}\,V(\lambda) = e^{-\rho}R^{-1}\sum_{w\in W}\mathrm{sgn}(w)\;w\bigl(e^{\lambda+\rho}S_\lambda\bigr).$$4

and the Brundan-Stroppel weight and cap diagrams. The atypical roots of a dominant integral highest weight $$\mathrm{ch}\,V(\lambda) = e^{-\rho}R^{-1}\sum_{w\in W}\mathrm{sgn}(w)\;w\bigl(e^{\lambda+\rho}S_\lambda\bigr).$$5 form a maximal orthogonal isotropic set $$\mathrm{ch}\,V(\lambda) = e^{-\rho}R^{-1}\sum_{w\in W}\mathrm{sgn}(w)\;w\bigl(e^{\lambda+\rho}S_\lambda\bigr).$$6. The weight diagram splits the $$\mathrm{ch}\,V(\lambda) = e^{-\rho}R^{-1}\sum_{w\in W}\mathrm{sgn}(w)\;w\bigl(e^{\lambda+\rho}S_\lambda\bigr).$$7 symbols into atypical components $$\mathrm{ch}\,V(\lambda) = e^{-\rho}R^{-1}\sum_{w\in W}\mathrm{sgn}(w)\;w\bigl(e^{\lambda+\rho}S_\lambda\bigr).$$8 of sizes $$\mathrm{ch}\,V(\lambda) = e^{-\rho}R^{-1}\sum_{w\in W}\mathrm{sgn}(w)\;w\bigl(e^{\lambda+\rho}S_\lambda\bigr).$$9, and $R$0 is piecewise disconnected if between consecutive clusters $R$1 and $R$2 there are at least $R$3 ordinary symbols (Chmutov et al., 2014).

For a PDC module, the main character formula is

$R$4

The summary then rewrites this as a nested sum by introducing the subgroup

$R$5

so that the Weyl-group sum decomposes into cosets $R$6 and an inner alternating sum over $R$7. Because $R$8 is a direct product of symmetric groups, the inner layer factorizes into $R$9 blockwise sums, one for each atypical component (Chmutov et al., 2014).

The PDC formula interpolates between two previously distinguished regimes. When $S_\lambda$0, the module is totally connected and the result reduces to the usual single-term Kac-Wakimoto formula after cancellation of the factor $S_\lambda$1. When $S_\lambda$2, the module is totally disconnected, $S_\lambda$3, and the result recovers the Bernstein-Leites formula. The nested structure therefore records precisely how atypical blocks interact: the outer Weyl sum moves clusters, while the inner sums resolve permutations inside each cluster (Chmutov et al., 2014).

6. Tokuyama deformations, ice models, and combinatorial pattern sums

A deformed Weyl-type formula of Tokuyama type for $S_\lambda$4 is given by

$S_\lambda$5

where $S_\lambda$6 is the $S_\lambda$7-deformed type $S_\lambda$8 Weyl denominator, $S_\lambda$9 is the doubling map

$\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$00

and the sum runs over strict type $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$01 Gelfand-Tsetlin patterns with top row $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$02 satisfying the parity condition that every generic entry is even. The statistics $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$03, $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$04, and $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$05 determine the weight factor

$\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$06

When $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$07, the formula reduces to the ordinary Weyl character formula; when $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$08, it recovers Tokuyama’s $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$09 case; and in the symplectic specialization it recovers the Hamel-King formula of type $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$10 (Friedberg et al., 2014).

The proof is itself nested. First, $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$11 is realized as the unramified Whittaker coefficient of a maximal-parabolic Eisenstein series on $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$12 via the Casselman-Shalika formula on the dual group $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$13. Second, that coefficient is rewritten as a crystal-sum over the metaplectic double cover of $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$14. The summary identifies a short-pattern induction as the nested combinatorial core of the argument and emphasizes an equality of sums over short patterns of type $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$15 and type $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$16 (Friedberg et al., 2014).

A broader family of deformations is obtained from free-fermion six-vertex models. In types $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$17, $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$18, $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$19, and $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$20, the partition functions $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$21 are expressed both as alternating Weyl sums with deformed denominator factors and as products of deformed denominators with ordinary characters. For example, in type $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$22,

$\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$23

and this is also

$\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$24

The six-vertex construction interprets the left-hand side as a partition function, proves divisibility by the deformed denominator via the Yang-Baxter equation, and shows that after the specialization $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$25 only one $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$26-vertex remains in each row-pair, leaving exactly the classical Weyl-character sum. The same source explicitly remarks that one may “nest” further by assigning each spectral line its own deformation parameter $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$27 rather than a single global $\sum_{w\in W}(-1)^{\ell(w)}w(\cdots)$28 (Brubaker et al., 2014).

These Tokuyama and lattice-model formulas show that nested Weyl-type expansions need not be limited to irreducible characters in their undeformed form. They also arise for deformed denominators, Whittaker coefficients, and partition functions, where the Weyl alternation survives but is reorganized through pattern statistics, short-pattern induction, or row-by-row spectral layering.