On the trace of Coxeter elements

G. Lusztig

REVIEW 2 major objections 5 minor 2 references

On the trace of Coxeter elements

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper proves that for any irreducible Weyl group, the trace of a Coxeter element on any irreducible complex representation is one of 0, 1, or -1, and extends this to a statement about Iwahori-Hecke algebras.

desk verdict A believable first written proof of Macdonald's trace claim plus a Hecke analogue; two verifiable proof gaps (D_n induction, exceptional table extraction) keep it from being fully polished. read the letter →

arxiv 2411.18495 v1 pith:FRYOYMP7 submitted 2024-11-27 math.RT

G. Lusztig This is my paper · ORCID

classification math.RT MSC 20F5520C1520C08

keywords CoxeterelementWeylgroupirreduciblerepresentationtraceIwahori-Heckealgebratwo-sidedcellsexteriorpowersMacdonaldconjecture

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

This paper proves a claim about Coxeter elements in Weyl groups that I. G. Macdonald stated in the early 1970s but left unpublished: for any irreducible Weyl group $W$, any Coxeter element $w$ (a product of the simple reflections in some order), and any irreducible complex representation $E$ of $W$, the trace $\operatorname{tr}(w,E)$ is one of the three numbers $0$, $1$, $-1$. The proof covers the classical families $A$, $B$, $C$, $D$ by direct counting arguments, and the exceptional Weyl groups $E_6$, $E_7$, $E_8$, $F_4$, $G_2$ by checking the known character tables. The paper also proves a Hecke-algebra refinement: in the Iwahori-Hecke algebra, the trace of the basis element $T_w$ on $E(v)$ is either $0$ or a single monomial $\pm v^{m_E}$, with $m_E$ a nonnegative integer. The noncrystallographic finite Coxeter groups ($H_3$, $H_4$, dihedral groups) do not satisfy the analogue, so the phenomenon is specific to Weyl groups.

What carries the argument

The load-bearing object is the Coxeter element $w$, the product of all simple reflections of $W$ in a fixed order; its order $h$ is the Coxeter number. The counting identity that carries the argument is the centralizer fact: the centralizer of $w$ in $W$ is cyclic of order $h$, so by the orthogonality of characters the squares of $\operatorname{tr}(w,E)$ over all irreducibles $E$ sum to $h$. If one can exhibit $h$ irreducibles with trace $\pm1$, all other traces must vanish. The paper exhibits those irreducibles via the exterior powers $\Lambda^i$ of the reflection representation, whose Coxeter traces are $\pm1$, together with further representations in types $B$ and $D$ built from symmetric-group representations (the symbols in [L84]). For the Hecke version, the key identity is $T_w^h = T_{w_0}^2$, valid for a suitable choice of $w$, together with the known involution on $E(v)$ that acts like a scalar times $v^{\nu-a_E+a_{E^!}}$ on $T_{w_0}$.

What would settle it

Independently compute $\operatorname{tr}(w,E)$ for every irreducible representation of $E_8$ (or $F_4$) using a direct construction of the representations and a Coxeter element, and check that every value lies in $\{0,1,-1\}$; the paper's 2.12 and 2.13 lists give the expected nonzero traces, so any coefficient other than $0$, $+1$, or $-1$ would disprove the claim.

Watch

Extended reading notes

Core claim

The central assertion, stated as 0.1(a), is that if $W$ is an irreducible Weyl group, $w$ is a Coxeter element, and $E$ is an irreducible complex representation of $W$, then $\operatorname{tr}(w,E)$ lies in $\{0,1,-1\}$. The argument splits according to type. For $S_n$, $B_n$, and $D_n$, the paper exhibits exactly $h$ irreducible representations with trace $\pm1$, where $h$ is the order of $w$, and then uses the fact that the centralizer of $w$ in $W$ is cyclic of order $h$: since the sum of the squares of all traces $\operatorname{tr}(w,E)$ equals $h$, any representation not among the exhibited ones must have trace $0$. For the exceptional types, the conclusion is read off the known character tables. The paper also proves the Iwahori-Hecke version 3.1(a): $\operatorname{tr}(T_w, E(v)) = \operatorname{tr}(w,E)\, v^{m_E}$ for some $m_E \in \mathbb{N}$, so the Hecke trace is zero or a signed monomial; the proof uses the known involution on $E(v)$ and the identity $T_w^h = T_{w_0}^2$, which forces eigenvalues of $T_w$ to be $h$-th roots of unity times a fixed power of $v$.

Load-bearing premise

The load-bearing premise is that the known character tables and the two-sided cell parametrization for the exceptional Weyl groups are complete and correct, and that the unstated induction behind 1.2(a) for type $D_n$ is valid; if any of these fails, the corresponding case of the theorem collapses.

Editorial extensions

If this is right

For every irreducible Weyl group, the $h$ characters with nonzero Coxeter trace are precisely the ones listed in Section 1; all others have trace $0$, giving an extremely sparse character-value distribution.
The nonzero traces align with the two-sided cell decomposition: contributions occur only on the cells $c_i$ attached to the exterior powers $\Lambda^i$ and on the exceptional cells, with the number of contributing irreducibles equal to $1, 2, 4, 6, 10$, or $2$ depending on the cell.
In the Iwahori-Hecke algebra, the element $T_w$ acts on each irreducible $E(v)$ with trace that is either zero or a single signed monomial $\pm v^{m_E}$, so there is no cancellation in that trace.
For a nonzero trace, the exponent $m_E$ is given explicitly by Proposition 3.3: if $E \in X_{c_i}$ then $m_E = 2r - 2i$, where $r$ is the rank of $W$.
The noncrystallographic analogue fails, so the $\{0,1,-1\}$ bound singles out Weyl groups among finite Coxeter groups.

Reading between the lines

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

The centralizer-counting method is robust enough that one could test it on other elements of Weyl groups, or on other finite reflection groups, by looking for a set of $h$ irreducibles with trace $\pm1$; the paper's negative results for $H_3$, $H_4$, and dihedral groups show where that search fails.
Because the trace is either $0$ or a sign, the Coxeter element acts as a kind of membership test: it distinguishes the cells that carry the 'Coxeter bit' in the cell decomposition from those that do not, a role reminiscent of unipotent representation theory.
The Hecke-algebra monomiality may have consequences for the zeros and poles of Kazhdan-Lusztig polynomials attached to Coxeter elements, though the paper does not develop this direction.
One could try to make the exceptional-type verification independent of the published tables by computing the traces directly from a computer construction of the representations; that would remove the reliance on table completeness.

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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

Short version: this is a clean, believable proof of Macdonald's old assertion, plus a Hecke algebra analogue. The main content is the counting lemma 0.2, which is elegant and correct, and the explicit trace computations for classical types. As a proof of 0.1(a) it is essentially complete modulo two gaps: the D_n case leans on 1.2(a), whose induction is only sketched, and the exceptional types are delegated to 'examining existing tables' without showing the extraction. Neither gap looks fatal—the theorem is very likely true, and both are verifiable—but as written they are real dents in completeness.

What's actually new: a first written proof of Macdonald's statement, and the Hecke algebra version 3.1(a), which is a genuinely useful strengthening. The argument around 3.2 using T_w^h = T_{w0}^2 plus [L84,5.12.2] is slick and gives the monomial form. Section 2's cell-by-cell decomposition is nice and ties the trace statement to cell theory; the explicit lists 2.7-2.14 are a service even if they come from tables. The noncrystallographic counterexample section is a good sanity check.

Soft spots in proportion: - 1.2(a) is asserted to follow by induction from 1.1 and restriction to S_{n-1}. That is plausible, but no details are given. The D_n construction of 'M_i depends on it, and the trace formula for 'M_i inherits the gap. This is the weakest spot in the classical part. - For exceptional types, Proposition 2.6(c) and the sums 2.10-2.14 are taken from existing tables. That is standard practice for finite type checks, and the tables are independent of the theorem, so not circular. But the paper does not show the individual traces or the completeness check, so a skeptical reader has to redo the table work to certify the case. That is an addressable presentation gap, not a mathematical error. - The Hecke argument assumes [L84,5.12.2] about the involution on E(v); that's an established result, and the trace specialization at v=1 correctly recovers 0.1(a). No circularity.

Overall: I trust the result. It deserves a serious referee; the proof is mostly there and the gaps are fillable. A referee should ask for more detail on 1.2(a) and a more explicit description of the exceptional-type table extraction, then accept.

Referee Report

2 major / 5 minor

Summary. The paper proves Macdonald's conjecture that for an irreducible complex representation E of a Weyl group W, the trace of a Coxeter element w on E belongs to {0,1,-1}. The proof uses a counting argument: if one can exhibit h distinct irreducible representations on which w has trace ±1, where h is the order of w and also the order of its centralizer, then the orthogonality relations force all other traces to vanish. The classical types A, B, and D are treated by explicit constructions from exterior powers and induction. The exceptional types are handled by asserting that certain cell sums, listed in §2.10–2.14, contain all contributing irreducibles. Section 3 proves an Iwahori-Hecke algebra analogue: tr(T_w,E(v)) equals tr(w,E) v^{m_E} for some nonnegative integer m_E, hence is either 0 or a monomial with coefficient ±1. Section 4 shows that the analogous statement fails for noncrystallographic Coxeter groups, with explicit counterexamples for H3, H4, and dihedral groups.

Significance. If the proof is completed, the paper settles a fifty-year-old unpublished conjecture of I.G. Macdonald in a clean and conceptual way. The counting reduction in §0.2 is elegant and rigorous, and the classical-type constructions are explicit and verifiable. The Hecke algebra analogue in §3 is a natural strengthening and the noncrystallographic counterexamples in §4 provide useful boundary information. The paper also draws a interesting connection to two-sided cells and special representations. The main gaps are completeness issues: the induction in §1.2 is only sketched and the exceptional-type verification in §2.6(c) is delegated to unnamed table inspections. These gaps are fillable, and the mathematical claims are very likely correct, but the written proof is not yet fully self-contained at these load-bearing points.

major comments (2)

[§1.2(a), used in §1.4] The proof of assertion (a) in §1.2 is a single sentence: 'This is proved by induction on n, using the results in 1.1 for S_{n−1} and the known results about restricting an E ∈ Irr(S_n) to S_{n−1}.' The first part, tr(w', Λ'_i) = (−1)^i, is load-bearing because §1.4 uses it to compute tr(w, M'_i) for type D_n. The induction step is not written out: one must show that the (n−1)-cycle acts on the specified Specht modules with the stated signs, and the branching argument is not immediate from §1.1 alone. The vanishing claim for all other irreducibles in §1.2(a) is also stated without proof. Please supply the full induction argument, including the branching rules, or give a precise reference that contains this exact statement.
[§2.6(c) and §2.10–2.14] For the exceptional Weyl groups, Proposition 2.6(c) asserts that no cells other than the c_i and the exceptional cells contribute to E(W), and the lists in §2.10–2.14 describe the corresponding cell sums. The proof says only that in the exceptional case 'this follows by examining the existing tables'. The paper does not tabulate the individual traces tr(w,E) or the cell u(E) for each irreducible used, and it does not identify the specific tables and labeling conventions that would allow a reader to verify the computation. Because the counting argument in §0.2 requires exactly h irreducible representations with trace ±1, the completeness of these lists is essential to the proof of 0.1(a) for E6, E7, E8, F4, and G2. This is a proof-completeness gap. Please provide a verifiable derivation: for instance, a full table of (E, u(E), tr(w,E)) for each exceptional group, or a reproducible computation using Chevie, GAP, or another package, together with a statement of the exact reference used for the character tables.

minor comments (5)

[Abstract and §0.1] There is a typo: 'Coxeter elememt' should be 'Coxeter element'.
[§2.5] The phrase 'free C-vector spacee' contains a typo; it should be 'vector space'.
[§2.16] In the line 'If Γ(c_i) = S_2 and W is simply laced then p_i = 0. (We use that 1/2 − 1/2 = 1.)', the arithmetic statement is false: 1/2 − 1/2 = 0, not 1. The intended identity is presumably '1/2 − 1/2 = 0', which matches the asserted value p_i = 0.
[§1.1] In the description of the partition for Λ^i, the notation '1 ≤ 1 ≤ 1 ≤ · · · ≤ 1 ≤ n − i' is ambiguous because the number of leading 1's is not specified. Writing the partition as (n−i, 1^i) would remove the ambiguity.
[§3.1] The assertion that tr(T_y,E(v)) ∈ C[v] for all y ∈ W is used without a reference. Please cite the precise result, for example the relevant statement in [KL] or [L84, Chapter 3].

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central trace theorem is proved by a self-contained classical counting argument plus independent character-table checks, and the self-citations do not assume the target result.

full rationale

The derivation chain for the central claim 0.1(a) is not circular. For type A, the paper computes tr(w, Λ^i) = (-1)^i on an n-cycle and then uses the trace-orthogonality argument 0.2, which is a self-contained identity following from the cyclic centralizer of w; no fitted parameter or prior trace theorem is invoked. The type B_n and D_n cases use the same reduction, exhibiting 2n or 2n-2 irreps with traces ±1 and then applying 0.2; the parametrization cited from [L84] is imported only as a labeling of irreps, not as the trace statement. The D_n case uses 1.2(a), which is itself proved by induction from the S_{n-1} case and standard branching rules; the induction is only sketched, but that is a proof-completeness gap, not a circular dependence. For exceptional Weyl groups, the paper explicitly relies on 'the known character tables' (0.1) and says in 2.6(c) that the cell decomposition statement 'follows by examining the existing tables'; these are external computations independent of the theorem being proved, so they are not circular inputs. Section 2's cell-theoretic description is organizational: it records which cells contribute, and the quoted table checks are again external. Section 3's Hecke-algebra version 3.1(a) is proved using 0.1(a) in the final specialization step, but since 0.1(a) has already been established, this is a legitimate corollary rather than an assumption. The cited [L84, 5.12.2] supplies an involution on Hecke modules, unrelated to the trace statement, and [L17] is used only to define 'exceptional cells.' No fitted value is renamed as a prediction, no uniqueness theorem is imported from the author's prior work to force a choice, and no ansatz is smuggled in via citation. The main theorem is very likely true; the noted gaps are verifiability gaps in the exceptional and D_n cases, not circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted and no new entities are introduced. The proof rests on standard facts about Coxeter elements, character orthogonality, known parametrizations of irreducible representations, existing character tables, and the cited theorem [L84,5.12.2]. The only genuine domain assumption is that the exceptional cases are disposed of by external table data.

assumptions (5)

standard math For an irreducible Weyl group W and a Coxeter element w of order h, the centralizer C_W(w) is cyclic of order h.
Invoked in 0.2 to obtain the character orthogonality count sum_E tr(w,E)^2 = h. If this fact failed, the counting argument that forces all other traces to zero would collapse.
standard math Character orthogonality: sum over Irr(W) of |tr(w,E)|^2 equals |C_W(w)|.
Used in 0.2 to conclude that once h irreps with trace plus or minus 1 are found, every remaining irrep has trace 0.
domain assumption The parametrization of Irr(W) by partitions and symbols from [L84,4.4-4.6], together with standard restriction rules for symmetric groups.
Used in 1.1-1.4 to identify the representations Lambda^i, tilde E_i, M_i, and primed M_i and to compute their traces. The paper cites [L84] instead of proving the parametrization.
domain assumption For exceptional Weyl groups, the known character tables correctly give all traces, including the cell data listed in 2.10-2.14.
The main statement for E6, E7, E8, F4, and G2 is asserted to follow by examining existing tables. The extraction is not shown in the paper, so the proof depends on the correctness and completeness of external table data.
standard math The theorem [L84,5.12.2]: on each Hecke module E(v) there is an involution sigma with sigma T_{w0} = T_{w0} sigma = v^{nu - a_E + a_{E!}}.
This is the key input in 3.2 that forces eigenvalues of T_w to be of the form zeta v^{m_E}, which is what yields the Hecke algebra version 3.1(a).

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Pith. "Pith review of On the trace of Coxeter elements." pith.science (2026). https://pith.science/paper/FRYOYMP7

@misc{pith2026241118495,
  author       = {Pith},
  title        = {Pith review of: On the trace of Coxeter elements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FRYOYMP7}},
  note         = {Machine review of arXiv:2411.18495}
}

read the original abstract

Let W be a Weyl group and let w be a Coxeter elememt of minimal length of W. In the early 1970's I.G.Macdonald stated that the trace of w on an irreducible representation of W is 0,1 or -1. In this paper we give a proof of this statement and of an Iwahori-Hecke algebra version of it.

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Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

53 (1979), 165-184

[KL] D.Kazhdan and G.Lusztig, Representations of Coxeter groups and Hecke algebras , Inv.Math. 53 (1979), 165-184. [L76] G.Lusztig, Coxeter orbits and eigenspaces of Frobenius , Invent.Math. 28 (1976), 101-

work page 1979
[1984]

475 (2017), 4-20

[L17] G.Lusztig, Exceptional representations of Weyl groups , J.Alg. 475 (2017), 4-20. Department of Mathematics, M.I.T., Cambridge, MA 02139

work page 2017

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

53 (1979), 165-184

[KL] D.Kazhdan and G.Lusztig, Representations of Coxeter groups and Hecke algebras , Inv.Math. 53 (1979), 165-184. [L76] G.Lusztig, Coxeter orbits and eigenspaces of Frobenius , Invent.Math. 28 (1976), 101-

work page 1979

[2] [1984]

475 (2017), 4-20

[L17] G.Lusztig, Exceptional representations of Weyl groups , J.Alg. 475 (2017), 4-20. Department of Mathematics, M.I.T., Cambridge, MA 02139

work page 2017