Visualizing the Support of Kostant's Weight Multiplicity Formula for the Rank Two Lie Algebras

Eduardo Torres Davila; Fabrice O. Ulysse; Gordon Rojas Kirby; Joseph Rennie; Juan Ramirez; Marissa Loving; Pamela E. Harris

REVIEW 3 major objections 4 minor 14 references

Visualizing the Support of Kostant's Weight Multiplicity Formula for the Rank Two Lie Algebras

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper completely determines the Weyl alternation sets $A(\lambda,\mu)$ for the four rank-two Lie algebras $\mathfrak{so}_5(\mathbb{C})$, $\mathfrak{sp}_4(\mathbb{C})$, $\mathfrak{so}_4(\mathbb{C})$, and $\mathfrak{g}_2$: each Weyl…

desk verdict Theorem 3.1 is false as stated: the {1,s1} case omits the shared condition J2, and a concrete counterexample shows the paper's central case lists cannot be trusted without recomputation. read the letter →

arxiv 1908.08405 v1 pith:24LTWSWO submitted 2019-08-22 math.CO math.RT

Pamela E. Harris , Marissa Loving , Juan Ramirez , Joseph Rennie , Gordon Rojas Kirby , Eduardo Torres Davila , Fabrice O. Ulysse This is my paper · ORCID

classification math.COmath.RT MSC 17B1005E10

keywords WeylalternationsetKostantweightmultiplicityformulapartitionfunctionranktwoLiealgebrasdiagramfundamentallatticeroot

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Using Kostant's weight multiplicity formula, the paper sets out to identify, for the four rank-two complex Lie algebras $\mathfrak{so}_4(\mathbb{C})$, $\mathfrak{so}_5(\mathbb{C})$, $\mathfrak{sp}_4(\mathbb{C})$, and $\mathfrak{g}_2$, exactly which Weyl group elements contribute a nonzero term to the multiplicity of a weight. Its central claim is that this set, the Weyl alternation set $A(\lambda,\mu)$, is completely described by a finite list of cases governed by linear inequalities in the coordinates of the highest weight $\lambda$ and the weight $\mu$, for every integral $\lambda$ and dominant integral $\mu$. This matters because the formula sums over the whole Weyl group while in practice most terms vanish, so knowing the support turns multiplicity computation into checking a handful of inequalities. The paper also draws Weyl alternation diagrams that visualize, on the fundamental weight lattice, which regions of weights share the same alternation set, including the regions where it is empty.

What carries the argument

The load-bearing object is the Weyl alternation set $A(\lambda,\mu) = \{\sigma \in W : \wp(\sigma(\lambda+\rho)-(\mu+\rho)) > 0\}$, where $\wp$ is Kostant's partition function, counting the number of ways a weight can be written as a nonnegative integral combination of positive roots. The mechanism that carries the argument is rewriting $\sigma(\lambda+\rho)-(\mu+\rho)$ in the simple-root basis $\alpha_1, \alpha_2$ for each Weyl group element, so that membership reduces to a pair of linear inequalities in the coordinates $c_1, c_2$ of $\lambda$ and $n, m$ of $\mu$; divisibility lemmas restrict the relevant lattice, and intersecting the resulting half-plane solutions yields the finite case lists. Plotting the boundary lines of these inequalities on the fundamental weight lattice produces the Weyl alternation diagrams whose empty regions the paper classifies geometrically.

What would settle it

Take $\mathfrak{so}_5(\mathbb{C})$, choose an integral $\lambda$ with $c_2$ even and a dominant integral $\mu$ with $m$ even, compute the two coefficients of $\sigma(\lambda+\rho)-(\mu+\rho)$ for all eight Weyl group elements, and compare the resulting set of contributing $\sigma$ with the subset predicted by Theorem 3.1; any pair whose true alternation set is not among the twenty-five listed cases, or appears in two of them, would refute the claimed completeness, and a random sweep over a large box of such pairs would settle the issue.

Watch

Extended reading notes

Core claim

For each of the Lie algebras $\mathfrak{so}_5(\mathbb{C})$ (type $B_2$), $\mathfrak{sp}_4(\mathbb{C})$ (type $C_2$), $\mathfrak{so}_4(\mathbb{C})$ (type $D_2$), and $\mathfrak{g}_2$ (type $G_2$), the paper establishes a complete, explicit description of the Weyl alternation set $A(\lambda,\mu)$. For every Weyl group element $\sigma$ it computes $\sigma(\lambda+\rho)-(\mu+\rho)$ in the simple-root basis, obtaining a pair of affine linear forms in the coordinates of $\lambda$ and $\mu$; $\sigma$ lies in $A(\lambda,\mu)$ exactly when both coordinates are nonnegative, which is the condition that Kostant's partition function counts at least one expression for that weight difference. Intersecting these membership conditions on the relevant lattice—with parity conditions on $c_2$ for $B_2$, on $c_1$ for $C_2$, on both for $D_2$, and none for $G_2$—produces the case lists of Theorems 3.1, 3.2, 3.3, and 3.4, each stating that $A(\lambda,\mu)$ is one of a finite collection of subsets of the Weyl group, the empty set included. The accompanying diagrams shade these regions on the fundamental weight lattice, giving a complete visual census of the support of Kostant's partition function in these four algebras.

Load-bearing premise

The classification rests on an enumeration step that is asserted but not shown in detail: that intersecting the individual membership conditions over all Weyl group elements produces exactly the listed alternation sets, with every valid combination of inequalities accounted for exactly once; if that enumeration is incomplete or overlapping, the complete descriptions of $A(\lambda,\mu)$ would be wrong even though the per-element conditions are correct.

Editorial extensions

If this is right

Weight-multiplicity computations for $\mathfrak{so}_5(\mathbb{C})$, $\mathfrak{sp}_4(\mathbb{C})$, $\mathfrak{so}_4(\mathbb{C})$, and $\mathfrak{g}_2$ can skip every Weyl group element excluded by the relevant linear inequalities, leaving only the terms that actually contribute to Kostant's formula.
The Weyl alternation diagrams give an at-a-glance census of which weight regions have an empty alternation set, with empty regions shaped like squares, stars, hexagons, or crosses whose size grows as $\mu$ grows.
The theorems extend the earlier $\mathfrak{sl}_3(\mathbb{C})$ lattice-pattern description to all four rank-two Lie algebras, covering arbitrary dominant $\mu$ rather than only the previously treated $\mu=0$ case.
The case lists provide the base pattern for the rank-three question posed in the paper, such as the Weyl alternation diagrams for $\mathfrak{sl}_4(\mathbb{C})$, where the same inequality-intersection method could be applied.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the enumeration is complete, then the empty regions in the diagrams are precisely the sets of weights $\lambda$ where all per-element inequality pairs conflict on the admissible lattice, and their square, star, and hexagon shapes are explained by which pairs of boundary lines intersect first—a structure the paper observes but does not fully formalize.
A natural testable extension is to implement the case lists and audit them computationally by brute-force evaluation of Kostant's partition function on random lattice points, which would settle the exhaustiveness question directly in rank two.
The same coordinate-expansion scheme should transfer to other rank-two root-system data, with the number of cases growing with the Weyl group order, and the rank-three question the paper poses for $\mathfrak{sl}_4(\mathbb{C})$ would reveal whether the method scales beyond dihedral Weyl groups.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: Theorem 3.1 is false as stated. The row for {1,s1} lists J1, J3, ¬J4, ¬J5, but both 1 and s1 require J2, and J2 is not implied by those four conditions. Example: λ=6ϖ2, μ=ϖ1+6ϖ2. Then J1=2≥0, J3=1≥0, J4=-8<0, J5=-10<0, so the row conditions hold. But J2=-1<0, so neither 1 nor s1 lies in A(λ,μ). Direct coordinate checks give A(λ,μ)=∅, not {1,s1}.

What the paper does well: the coefficient computations for each Weyl group element are explicit and checkable, and the extension from μ=0 to general dominant μ is new relative to the cited literature. The diagrams are a nice visual aid, and the exposition is accessible. The method itself is fine: reduce membership to linear inequalities and intersect.

The problem is that the intersection step is done by hand and is not shown. The proof says “intersecting these solution sets produces the desired results” without giving the actual intersection analysis. The counterexample above shows this is not a minor typo; the enumeration itself is wrong. The same hand-intersection method is used in Theorems 3.2–3.4, so those case lists carry the same risk and need independent verification. Section 4's informal if-and-only-if statements about the shapes of empty regions inherit the same fragility. The abstract also overstates the domain: the theorems restrict μ to dominant integral weights, not all pairs.

This paper is for people who want to simplify Kostant's multiplicity formula in rank 2 and for undergraduate research. At present it is not a reliable reference. A corrected version, with the case lists recomputed by computer or presented as a full intersection table, would be genuinely useful.

My recommendation: send it to a referee, but with a clear instruction that the case lists must be verified computationally. The question is worthwhile and the method is transparent, but the current version's central claims are not correct.

Referee Report

3 major / 4 minor

Summary. The paper aims to give complete case-by-case descriptions of the Weyl alternation set A(λ,μ) for the rank two Lie algebras so4(C), so5(C), sp4(C), and g2, for every integral λ and dominant integral μ. Theorems 3.1–3.4 list subsets of the Weyl group according to which of a set of linear inequalities in the coordinates of λ and μ hold, with the proofs computing for each Weyl group element the two inequalities equivalent to membership in A(λ,μ) and then asserting that intersecting these solution sets yields the tables. Section 4 uses these lists to draw Weyl alternation diagrams for various μ and claims precise shapes (square, hexagon, star, strip, cross) for the empty region. The paper is not correct as written: the row {1,s1} in Theorem 3.1 omits the inequality J2 that the proof itself shows is necessary for both 1 and s1 to lie in A(λ,μ), and an explicit counterexample satisfies all stated row conditions while A(λ,μ)=∅.

Significance. If correct, the paper would reduce the computation of the support of Kostant's partition function to checking linear inequalities for the four rank two algebras, and it would provide a visual census of the empty and nonempty regions extending earlier work on sl3(C). The paper has real strengths: the individual membership computations for each Weyl group element are explicit and checkable, the parity conditions for the root lattice are organized cleanly, and the geometric presentation of the inequalities is useful. However, the central claim, namely the correctness and completeness of the case lists in Theorems 3.1–3.4, is false. Since the Weyl alternation diagrams in Section 4 are colored according to exactly those case lists, the main contribution cannot be accepted as stated.

major comments (3)

[§3.1, Theorem 3.1] The row {1,s1} lists the conditions J1, J3, ¬J4, and ¬J5, but omits J2, which the proof itself shows is necessary for both 1 and s1 to belong to A(λ,μ). For λ=6ϖ2 and μ=ϖ1+6ϖ2, i.e., (c1,c2)=(0,6) and (n,m)=(1,6), the parity assumptions are satisfied and the stated row conditions hold: J1=2≥0, J3=1≥0, J4=−8<0, and J5=−10<0. However, J2=c1+c2−n−m=−1<0, so neither 1 nor s1 satisfies the membership inequalities derived in the proof. In fact, all eight coefficient pairs are negative in this example, so A(λ,μ)=∅, not {1,s1}. This is a direct numerical counterexample to Theorem 3.1 and therefore to the paper's central claim.
[§3.1–§3.4, proofs of Theorems 3.1–3.4] The error is not an isolated typo but a systematic failure of the asserted intersection step. In Theorem 3.1, row {1,s2} lists J2,J4,¬J3,¬J6 but omits J1, which is necessary for both 1 and s2; row {s1,s2s1} lists J2,J5,¬J1,¬J7 but omits J3, which is necessary for both s1 and s2s1; and row {1,s1,s2s1} lists J3,J4,¬J5,¬J6 even though the proof shows s2s1∈A(λ,μ) only if J5 holds, so the stated conditions actually force s2s1 to be absent. Each proof in Theorems 3.1–3.4 closes with the sentence 'intersecting these solution sets produces the desired results' without performing the intersection analysis, and the counterexample above shows the omitted analysis is not merely unproven but incorrect. Since the diagrams in Section 4 are colored from these same case lists, the diagrams inherit the error.
[§4, especially §4.1.3 and Figures 13a–13c] The 'if and only if' statements describing when the empty region is a square with an edge on top, a square with a vertex up, or an 8-pointed star are asserted from the diagrams rather than derived from the inequalities and the lattice parity conditions. This is a secondary gap in the exposition, but it matters because the diagrams are constructed from the incorrect case lists; after Theorem 3.1 is corrected, the shape classifications must be re-examined. The claim that changing μ only translates the solution sets also requires care because the parity of n and m changes which sublattice is plotted.

minor comments (4)

[§3.3, proof of Theorem 3.3] The displayed equivalence for 1∈A(λ,μ) contains 'c1−m/2 ≥0' where the second inequality should be 'c2−m/2 ≥0', and the same substitution is needed in the displayed equivalence for s1∈A(λ,μ).
[§4.1–§4.2] The text in Section 4.1 says 'we plot the conditions in Table 1' and Section 4.2 says 'Table 2'; these should refer to the condition tables (Table 5 and Table 6, respectively) rather than the Weyl group action tables.
[§3.3] The statement of Theorem 3.3 fixes n,m∈2N, but the proof begins with 'n,m∈N'; the parity assumptions should be stated consistently throughout.
[§3.4] Theorem 3.4 is typeset as a long sequence of separate display equations, which makes it difficult to verify exhaustiveness and disjointness; a single aligned case environment would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Weyl alternation set computations are direct expansions of the defining partition-function positivity condition, with no fitted parameters, no prediction-from-fit, and no load-bearing self-citation.

full rationale

The paper's central claims are the case lists in Theorems 3.1-3.4, which describe A(lambda, mu) for the rank-two algebras. The derivation chain is fully self-contained: the proofs expand sigma(lambda+rho)-(mu+rho) in the simple-root basis for each Weyl group element, then invoke Definition 2.1, which defines A(lambda, mu) as the set of sigma with both coefficients nonnegative. The displayed formulas in each proof, e.g. in Theorem 3.1, directly give the equivalence between membership and the inequalities J1-J8. The closing sentence 'intersecting these solution sets ... produces the desired results' is a computational step on the already-derived inequalities, not an appeal to an external or imported conclusion. There are no fitted parameters, no subset of data used to predict a closely related quantity, and no uniqueness theorem imported from prior work to force a choice. The citations to [4] and [8] are used only for definitions, diagrams, and background on Weyl alternation diagrams, not as evidence for the membership computations. Any issue with the claimed case lists would be a mathematical correctness problem, not a circularity problem; indeed the reviewer's own counterexample concerns the truth of the intersection, not the derivation depending on its conclusion. The paper is therefore self-contained against external definitions and merits a circularity score of 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters or invented entities appear. The central claim relies on standard Lie theory and on an implicit equivalence between positivity of Kostant's partition function and nonnegativity of simple-root coefficients, which is justified because the simple roots are themselves positive roots in rank two.

assumptions (4)

standard math Kostant's weight multiplicity formula, equation (1.1), gives multiplicities as an alternating sum over the Weyl group involving Kostant's partition function.
Invoked in Section 1 as the definition of the multiplicity being studied.
standard math The root system and Weyl group action data for B2, C2, D2, and G2 are as listed in Section 2 and Tables 1 through 4.
These action tables are used in every computation in Section 3; they are standard textbook data stated without proof.
domain assumption For rank two, positivity of Kostant's partition function is equivalent to the argument having nonnegative integer coefficients in the simple-root basis.
Used implicitly throughout Section 3 when converting partition-function positivity into pairs of inequalities; true because simple roots are themselves positive roots, but not stated explicitly.
domain assumption Type D2, so4(C), is treated in the same framework even though the introduction calls the Lie algebras simple and so4(C) is semisimple rather than simple.
The paper does not discuss the non-simplicity of so4; if the formula were not assumed to carry over to semisimple algebras, Theorem 3.3 would lack justification.

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Pith. "Pith review of Visualizing the Support of Kostant's Weight Multiplicity Formula for the Rank Two Lie Algebras." pith.science (2026). https://pith.science/paper/24LTWSWO

@misc{pith2026190808405,
  author       = {Pith},
  title        = {Pith review of: Visualizing the Support of Kostant's Weight Multiplicity Formula for the Rank Two Lie Algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/24LTWSWO}},
  note         = {Machine review of arXiv:1908.08405}
}

read the original abstract

The multiplicity of a weight in a finite-dimensional irreducible representation of a simple Lie algebra g can be computed via Kostant's weight multiplicity formula. This formula consists of an alternating sum over the Weyl group (a finite group) and involves a partition function known as Kostant's partition function. Motivated by the observation that, in practice, most terms in the sum are zero, our main results describe the elements of the Weyl alternation sets. The Weyl alternation sets are subsets of the Weyl group which contributes nontrivially to the multiplicity of a weight in a highest weight representation of the Lie algebras so_4(C), so_5(C), sp_4(C), and the exceptional Lie algebra g_2. By taking a geometric approach, we extend the work of Harris, Lescinsky, and Mabie on sl_3(C), to provide visualizations of these Weyl alternation sets for all pairs of integral weights \lambda and \mu of the Lie algebras considered.

Figures

Figures reproduced from arXiv: 1908.08405 by the authors.

**Figure 1.** Weyl alternation diagram for sl3(C) with µ = 0. Reproduced from [8]. More recently, Harris, Lescinsky, and Mabie considered the Lie algebra sl3(C) and took a geometric approach to determining the elements of the Weyl group that contribute nontrivially to m(λ, µ) by varying the weights λ and µ over the fundamental weight lattice [8]. This involved computing the Weyl alternation diagram (associated to the weight µ), … view at source ↗

**Figure 2.** Weyl alternation diagrams for the Lie algebras of type B2, C2, D2, and G2, where µ = 0. Reproduced from [4]. 2. Background In this section we provide the background necessary to make our approach precise. We use the notation and definitions of [3] and for the technical background on the representation theory of Lie algebras, as it relates to computing weight multiplicities, we refer the interested reader to [5]. For… view at source ↗

**Figure 3.** The root system of B2. s1 is the reflection through the hyperplane perpendicular to α1, and s2 is the reflection through the hyperplane perpendicular to α2 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (29 more)

**Figure 4.** Figure 4: The root system of C2. The Weyl group is generated by s1 and s2, where s1 is the reflection through the hyperplane perpendicular to α1, and s2 is the reflection through the hyperplane perpendicular to α2 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: The root system of D2. on the simple roots. Moreover, the action of the reflections on the simple roots and fundamental weights is given by si(αj) = ⎧⎪⎪ ⎨ ⎪⎪⎩ −αj if i = j αj if i ≠ j and si($j) = ⎧⎪⎪ ⎨ ⎪⎪⎩ $j − αj if i = j $j if i ≠ j. σ ∈ W 1 s1 s2 s2s1 σ(α1) α1 −α1 …

**Figure 6.** Figure 6: The root system of G2. σ ∈ W 1 s1 s2 s2s1 s1s2 s1s2s1 s2s1s2 (s2s1) 2 (s1s2) 2 s1(s2s1) 2 s2(s1s2) 2 (s2s1) 3 σ(α1) α1 -α1 β1 -β1 β3 -β3 β3 -β3 β1 −β1 α1 −α1 σ(α2) α2 β4 -α2 β2 -β4 β2 -β2 β4 -β2 α2 −β4 −α2 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: The fundamental weight lattice of Lie algebras of type B2, C2, D2, and G2. The construction of Weyl alternation diagrams entails graphing the corresponding linear inequalities found in Section 3 on the fundamental weight lattice of each respective Lie algebra we have …

**Figure 8.** Figure 8: Solution sets to linear inequalities corresponding to the Lie algebra of type B2. 4.1.1. Case µ = n$1. Figures 9a-9d illustrate the Weyl alternation diagrams for µ = n$1 such that n = 1, 2, 3, 4. We observe that the empty region takes the shape of a square oriented so …

**Figure 9.** Figure 9: Weyl alternation diagrams for the Lie algebra of type B2 with µ = n$1. 4.1.2. Case µ = m$2. Figures 10a-10d illustrate the Weyl alternation diagrams for µ = m$2 such that m = 2, 4, 6, 8. We observe that the empty region is in the shape of a square oriented so that an e…

**Figure 10.** Figure 10: Weyl alternation diagrams for the Lie algebra of type B2 with µ = m$2. 4.1.3. Case µ = n$1 + m$2. Figures 11a-11d illustrate the Weyl alternation diagrams for µ = n$1 + m$2. We observe that the empty region is in the shape of an 8-pointed star. Additionally, as n and …

**Figure 11.** Figure 11: Weyl alternation diagrams for the Lie algebra of type B2 with µ = n$1 + m$2. J2 J3 J5 J7 J8 J6 J4 J1 [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]

**Figure 12.** Figure 12: Set of linear inequalities determining the boundaries of the Weyl alternation sets for the Lie algebra of type B2. To explain the shapes that form in the empty region of each Weyl alternation diagram for the Lie algebra of type B2 we turn to [PITH_FULL_IMAGE:figures…

**Figure 13.** Figure 13: Different formations of the empty region for the Lie algebra of type B2. From Figure 13a, observe that the empty region becomes a square with an edge on top if and only if inequalities J1 and J6 intersect at or below J4 and also the inequalities J2 and J4 intersect st…

**Figure 14.** Figure 14: Solution sets to linear inequalities corresponding to the Lie algebra of type C2. 4.2.1. Case µ = n$1. Figures 15a-15d illustrate the Weyl alternation diagrams for µ = n$1 such that n = 2, 4, 6, 8. We observe that the empty region is in the shape of a square with an e…

**Figure 15.** Figure 15: Weyl alternation diagrams for the Lie algebra of type C2 with µ = n$1. 4.2.2. Case µ = m$2. Figures 16a-16d illustrate the Weyl alternation diagrams for µ = m$2 such that m = 1, 2, 3, 4. We observe that the empty region is in the shape of a square with a vertex on top…

**Figure 16.** Figure 16: Weyl alternation diagrams for the Lie algebra of type C2 with µ = m$2. 4.2.3. Case µ = n$1 + m$2. Figures 17a-17d illustrate the Weyl alternation diagrams for µ = n$1 + m$2 such that µ is a positive integral linear combination of the fundamental weights. We observe th…

**Figure 17.** Figure 17: Weyl alternation diagrams for the Lie algebra of type C2 with µ = n$1 + m$2. L1 L4 L6 L8 L7 L5 L3 L2 [PITH_FULL_IMAGE:figures/full_fig_p019_17.png]

**Figure 18.** Figure 18: Set of linear inequalities determining the boundaries of the Weyl alternation sets for the Lie algebra of type C2. To explain the shapes that form in the empty region of each Weyl alternation diagram for the Lie algebra of type C2 we turn to [PITH_FULL_IMAGE:figures…

**Figure 19.** Figure 19: Different formations of the empty region for the Lie algebra of type C2. From Figure 19a, observe that the empty region becomes a square with an edge on top if and only if the inequalities L2 and L5 intersect at or below L3 and the inequalities L1 and L3 intersect str…

**Figure 20.** Figure 20: Solution sets to linear inequalities corresponding to the Lie algebra of type D2. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_20.png]

**Figure 21.** Figure 21: Weyl alternation diagrams for the Lie algebra of type D2 with µ = n$1. 4.3.2. Case µ = m$2. Figures 22a-22d illustrate the Weyl alternation diagrams for µ = m$2 such that m = 2, 4, 6, 8. We observe that the empty region is a horizontal rectangle that stretches out inf…

**Figure 22.** Figure 22: Weyl alternation diagrams for the Lie algebra of type D2 with µ = m$2. 4.3.3. Case µ = n$1 + m$2. Figures 23a-23d illustrate the Weyl alternation diagrams for µ = n$1 + m$2. We observe that the empty region is in the shape of a cross that extends infinitely. 1 2 (a) µ…

**Figure 23.** Figure 23: Weyl alternation diagrams for the Lie algebra of type D2 with µ = n$1 + m$2. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_23.png]

**Figure 24.** Figure 24: Set of linear inequalities for determining the boundaries of the Weyl alternation sets for the Lie algebra of type D2. To explain the shapes that form in the empty region of each Weyl alternation diagram for the Lie algebra of type D2 we turn to [PITH_FULL_IMAGE:figu…

**Figure 25.** Figure 25: Different formation of the empty region for the Lie algebra of type D2. From Figure 25a, observe that the empty region becomes a vertical strip when the inequalities −c1−n−2 2 ≥ 0 and c1−n 2 ≥ 0 do not intersect and the inequalities c2−m 2 ≥ 0 and −c2−m−2 2 ≥ 0 inters…

**Figure 26.** Figure 26: Solution sets to linear inequalities corresponding to g2. In [PITH_FULL_IMAGE:figures/full_fig_p023_26.png]

**Figure 27.** Figure 27: Weyl alternation diagrams of g2 with µ = nα1. 4.4.2. Case µ = mα2. Figures 28a-28d give a geometric representation of the Weyl alternation diagrams for µ = mα2 with m = 1, 2, 3, 4. Observe that the empty region takes a hexagonal shape with an edge on top. We also note…

**Figure 28.** Figure 28: Weyl alternation diagrams of g2 with µ = mα2. 4.4.3. Case µ = nα1 + mα2. First, we assume that n and m satisfy the inequalities 2n + 1 > 3m and 2m + 1 ≤ n or 2n + 1 ≤ 3m and 2m + 1 > n. Figures 29a-29d give a geometric representation of the Weyl alternation diagrams. …

**Figure 29.** Figure 29: Weyl alternation diagrams of g2 where µ = nα1 +mα2 with 2n+1 > 3m and 2m + 1 ≤ n or 2n + 1 ≤ 3m and 2m + 1 > n. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_29.png]

**Figure 30.** Figure 30: Weyl alternation diagrams of g2 where µ = nα1+mα2 where 2n+1 > 3m and 2m + 1 > n. K1 K2 K4 K3 K10 K11 K5 K6 K12 K7 K8 K9 [PITH_FULL_IMAGE:figures/full_fig_p025_30.png]

**Figure 31.** Figure 31: Set of linear inequalities for determining the boundaries of the Weyl alternation sets for the Lie algebra of type G2. To explain the shapes that form in the empty region of each Weyl alternation diagram for the Lie algebra of type G2 we turn to [PITH_FULL_IMAGE:figu…

**Figure 32.** Figure 32: Different formations of the center empty region for the Lie algebra of type G2. From Figure 32a, we note that the hexagon with an edge on top will occur when the inequalities K1 and K3 intersect at or below K2 and the inequalities K4 and K2 intersect strictly above K1…

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P. E. Harris, M. Rahmoeller, L. Schneider, and A. Simpson. When is the q -multiplicity of a weight equal to a power of q ? https://arxiv.org/pdf/1905.10319.pdf . Preprint

1905 arXiv
[14]

V. S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations, Springer (1984)

1984

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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