Affine Coxeter Polynomial Overview

• Affine Coxeter polynomial is defined as the characteristic polynomial of a Coxeter transformation acting on the real root space modulo the null root.
• It is derived through determinant identities using Chebyshev polynomials, linking affine Cartan and adjacency matrices in untwisted settings.
• The polynomial factors into cyclotomic components, encoding affine exponents and Coxeter numbers that reflect the spectral structure of affine Lie algebras.

Searching arXiv for relevant papers on affine Coxeter polynomials and related conventions. Affine Coxeter polynomial denotes the characteristic polynomial associated with a Coxeter transformation for an affine Lie algebra or affine Coxeter system. In the untwisted affine setting, one begins with an affine Cartan matrix $C$, the corresponding simple reflections $\sigma_0,\dots,\sigma_n$, and a Coxeter transformation $w=\sigma_{\iota(0)}\cdots \sigma_{\iota(n)}$ determined by an ordering $\iota$. In the formulation of Damianou and Evripidou, the polynomial $f(x)=\det(xI_{n+1}-w)$, acting on the real root space modulo the null root $\delta$, is the affine Coxeter polynomial; it is related to the affine Cartan and adjacency matrices through determinant identities expressed via Chebyshev polynomials (Damianou et al., 2014). The same source computes explicit cyclotomic factorizations and affine exponents for all untwisted affine types. A separate, later treatment uses the term “characteristic polynomial” for affine Coxeter groups in a representation-theoretic setting and writes related closed forms by type, which makes clear that terminology and normalization vary across the literature (Feng et al., 16 May 2025).

1. Definition and algebraic setting

Let $\mathfrak g$ be an untwisted affine Kac–Moody algebra of type $\widetilde X_n$ with affine Cartan matrix $C$. Writing

$A=2I_{n+1}-C,$

Damianou–Evripidou introduce

$\sigma_0,\dots,\sigma_n$0

If the Dynkin graph is bipartite, then the affine Coxeter polynomial $\sigma_0,\dots,\sigma_n$1 satisfies

$\sigma_0,\dots,\sigma_n$2

Equivalently, $\sigma_0,\dots,\sigma_n$3 is the characteristic polynomial of a Coxeter transformation acting on

$\sigma_0,\dots,\sigma_n$4

where $\sigma_0,\dots,\sigma_n$5 is the null root (Damianou et al., 2014).

A choice of ordering $\sigma_0,\dots,\sigma_n$6 gives

$\sigma_0,\dots,\sigma_n$7

Its characteristic polynomial

$\sigma_0,\dots,\sigma_n$8

is called a Coxeter polynomial of the affine algebra. In all untwisted cases except $\sigma_0,\dots,\sigma_n$9, it is independent of $w=\sigma_{\iota(0)}\cdots \sigma_{\iota(n)}$0 (Damianou et al., 2014). This dependence issue is central: in affine type $w=\sigma_{\iota(0)}\cdots \sigma_{\iota(n)}$1, distinct conjugacy classes of Coxeter transformations occur, so the phrase “the affine Coxeter polynomial” may require a convention specifying the ordering or conjugacy class.

A distinct convention appears in a later representation-theoretic exposition of irreducible affine Coxeter groups, where one fixes an ordering of affine simple reflections $w=\sigma_{\iota(0)}\cdots \sigma_{\iota(n)}$2, forms $w=\sigma_{\iota(0)}\cdots \sigma_{\iota(n)}$3, and defines

$w=\sigma_{\iota(0)}\cdots \sigma_{\iota(n)}$4

in a reflection representation $w=\sigma_{\iota(0)}\cdots \sigma_{\iota(n)}$5 (Feng et al., 16 May 2025). This suggests that some apparent formula discrepancies in the literature arise from the choice of representation, normalization, or whether the null-root direction has been modded out.

2. Relation to affine Cartan and adjacency matrices

The determinant formalism links affine Coxeter polynomials to the spectra of affine Cartan and adjacency matrices. For an $w=\sigma_{\iota(0)}\cdots \sigma_{\iota(n)}$6 affine Cartan matrix $w=\sigma_{\iota(0)}\cdots \sigma_{\iota(n)}$7, the associated matrix

$w=\sigma_{\iota(0)}\cdots \sigma_{\iota(n)}$8

plays the role of a graph adjacency matrix or Cartan–adjacency matrix, and the polynomials $w=\sigma_{\iota(0)}\cdots \sigma_{\iota(n)}$9 are extracted from $\iota$0 by elementary substitutions (Damianou et al., 2014).

When the affine Dynkin graph is bipartite, the identity

$\iota$1

provides a direct passage from the determinant of a matrix built from $\iota$2 to the Coxeter polynomial. This construction is not merely formal: it yields closed forms for $\iota$3, hence for $\iota$4 and $\iota$5, and thereby for $\iota$6 itself in the bipartite cases (Damianou et al., 2014).

The same general theme reappears in work on weak order. Kim and Yun study the characteristic polynomial of the weak order on classical and affine Coxeter groups, defining for infinite groups such as irreducible affine types the modified characteristic polynomial

$\iota$7

and derive affine recurrences by decomposing subsets of the affine Coxeter graph into finite-type contributions (Kim et al., 2022). Although this is a different polynomial from the affine Coxeter polynomial of Damianou–Evripidou, both theories exploit the structure of affine Dynkin diagrams and the passage from affine to finite type by deleting the affine node. This suggests a broader methodological pattern: affine polynomial invariants often reduce to finite-type data plus an affine correction term.

3. Chebyshev-polynomial derivation

A defining feature of the Damianou–Evripidou analysis is the use of Chebyshev polynomials of the first and second kind. They recall the recurrences

$\iota$8

$\iota$9

together with

$f(x)=\det(xI_{n+1}-w)$0

By expanding $f(x)=\det(xI_{n+1}-w)$1, they obtain closed forms for the classical untwisted affine types (Damianou et al., 2014): $f(x)=\det(xI_{n+1}-w)$2

$f(x)=\det(xI_{n+1}-w)$3

$f(x)=\det(xI_{n+1}-w)$4

$f(x)=\det(xI_{n+1}-w)$5

These formulas are structurally significant because the trigonometric description of $f(x)=\det(xI_{n+1}-w)$6 immediately converts matrix determinants into roots on the unit circle. Since affine Coxeter polynomial roots are roots of unity in the untwisted cases under consideration, the Chebyshev framework serves as a bridge from matrix-theoretic data to cyclotomic factorization (Damianou et al., 2014).

For the exceptional untwisted affine types, the final polynomials are given explicitly rather than through the same family of determinant identities. The approach remains uniform at the level of outcome: closed-form expressions, factorization into cyclotomic factors, and recovery of affine exponents and Coxeter numbers (Damianou et al., 2014).

4. Cyclotomic factorization by affine type

A central theorem is that the roots of an affine Coxeter polynomial are roots of unity, so $f(x)=\det(xI_{n+1}-w)$7 factors into cyclotomic polynomials $f(x)=\det(xI_{n+1}-w)$8 (Damianou et al., 2014). For the standard bipartite affine Coxeter polynomials, the factorizations are as follows.

Type	Affine Coxeter polynomial $f(x)=\det(xI_{n+1}-w)$9	Cyclotomic factorization
$\delta$0	$\delta$1	$\delta$2
$\delta$3	$\delta$4	$\delta$5
$\delta$6	$\delta$7	$\delta$8
$\delta$9	$\mathfrak g$0	$\mathfrak g$1
$\mathfrak g$2	$\mathfrak g$3	$\mathfrak g$4
$\mathfrak g$5	$\mathfrak g$6	$\mathfrak g$7

For the exceptional $\mathfrak g$8-types, Damianou–Evripidou give the explicit polynomials

$\mathfrak g$9

with factorization

$\widetilde X_n$0

$\widetilde X_n$1

with factorization

$\widetilde X_n$2

and

$\widetilde X_n$3

described as a product over cyclotomic factors indexed by divisors associated with $\widetilde X_n$4 (Damianou et al., 2014).

These factorizations encode the eigenvalue structure of the Coxeter transformation and expose the arithmetic nature of affine Coxeter spectra. They also make transparent the multiplicity of the factor $\widetilde X_n$5, which reflects the inclusion of the affine exponent $\widetilde X_n$6 and, in several types, additional degeneracies forced by the affine structure.

5. Affine exponents and affine Coxeter numbers

The affine exponents $\widetilde X_n$7 and affine Coxeter number $\widetilde X_n$8 are characterized by the root formula

$\widetilde X_n$9

with

$C$0

In the standard untwisted bipartite case, Damianou–Evripidou recover the following data (Damianou et al., 2014).

Type	Affine exponents	Affine Coxeter number
$C$1	$C$2	$C$3
$C$4	$C$5	$C$6
$C$7	$C$8	$C$9
$A=2I_{n+1}-C,$0	$A=2I_{n+1}-C,$1	$A=2I_{n+1}-C,$2
$A=2I_{n+1}-C,$3	$A=2I_{n+1}-C,$4	$A=2I_{n+1}-C,$5
$A=2I_{n+1}-C,$6	$A=2I_{n+1}-C,$7	$A=2I_{n+1}-C,$8
$A=2I_{n+1}-C,$9	$\sigma_0,\dots,\sigma_n$00	$\sigma_0,\dots,\sigma_n$01
$\sigma_0,\dots,\sigma_n$02	$\sigma_0,\dots,\sigma_n$03	$\sigma_0,\dots,\sigma_n$04
$\sigma_0,\dots,\sigma_n$05	$\sigma_0,\dots,\sigma_n$06	$\sigma_0,\dots,\sigma_n$07

The same source states three ways to recover these parameters: directly from the factorization $\sigma_0,\dots,\sigma_n$08, from the spectrum of the affine Cartan or adjacency matrix via the Chebyshev determinant formulas, and through Steinberg’s “delete the branch-node” result (Damianou et al., 2014). In the last method, deleting the affine node yields a finite root system whose Coxeter polynomial $\sigma_0,\dots,\sigma_n$09 satisfies

$\sigma_0,\dots,\sigma_n$10

This creates a direct bridge between affine and finite exponents.

The later representation-theoretic exposition of affine Coxeter groups adopts the “extended exponents” viewpoint, explicitly adjoining $\sigma_0,\dots,\sigma_n$11 to the finite-type exponents $\sigma_0,\dots,\sigma_n$12 and then writing the spectrum of the affine Coxeter element as $\sigma_0,\dots,\sigma_n$13 (Feng et al., 16 May 2025). This is compatible at the level of exponents and Coxeter numbers with the untwisted affine lists above, even when polynomial formulas are presented in a different normalization.

6. The case $\sigma_0,\dots,\sigma_n$14 and convention-dependent formulas

The affine type $\sigma_0,\dots,\sigma_n$15 is the most delicate case because, unlike the other untwisted affine types, the Coxeter polynomial depends on the conjugacy class determined by the ordering of simple reflections (Damianou et al., 2014). For the standard bipartite ordering, the adjacency matrix is the cycle graph on $\sigma_0,\dots,\sigma_n$16 vertices, and one computes

$\sigma_0,\dots,\sigma_n$17

Thus

$\sigma_0,\dots,\sigma_n$18

and the resulting affine Coxeter polynomial is

$\sigma_0,\dots,\sigma_n$19

Its roots are

$\sigma_0,\dots,\sigma_n$20

so the affine exponents are $\sigma_0,\dots,\sigma_n$21 and the affine Coxeter number is $\sigma_0,\dots,\sigma_n$22 (Damianou et al., 2014).

By contrast, a later account of affine Coxeter groups states for $\sigma_0,\dots,\sigma_n$23 that

$\sigma_0,\dots,\sigma_n$24

in its chosen reflection-representation convention (Feng et al., 16 May 2025). The same source gives, for example, $\sigma_0,\dots,\sigma_n$25 (Feng et al., 16 May 2025). This difference should not be conflated with an arithmetic contradiction. Rather, the two formulas arise in distinct setups: one is the affine Coxeter polynomial attached to a Coxeter transformation on the real root space modulo the null root, with attention to conjugacy classes in $\sigma_0,\dots,\sigma_n$26; the other is a characteristic polynomial in a representation-theoretic affine-group formalism. A plausible implication is that “affine Coxeter polynomial” is not fully standardized across all subliteratures, especially in affine type $\sigma_0,\dots,\sigma_n$27.

This convention sensitivity is important in both structural and computational work. In any use of the polynomial, one must specify at least the affine type, the ordering or conjugacy class when relevant, and the representation or quotient space on which the Coxeter transformation acts.

7. Related polynomial invariants and broader context

The affine Coxeter polynomial belongs to a larger ecosystem of polynomial invariants attached to Coxeter systems. Kim–Yun study characteristic polynomials of weak order on classical and affine Coxeter groups, proving product factorizations for finite intervals and affine recurrences derived from separating subsets that omit or contain the affine node (Kim et al., 2022). Their affine formula in type $\sigma_0,\dots,\sigma_n$28,

$\sigma_0,\dots,\sigma_n$29

belongs to Möbius-theoretic poset combinatorics rather than Coxeter-element spectral theory, but it again exhibits the characteristic affine-to-finite reduction (Kim et al., 2022).

Feng–Liu–Wang, in turn, use a generating-set formalism for affine Coxeter groups $\sigma_0,\dots,\sigma_n$30 and define

$\sigma_0,\dots,\sigma_n$31

for finite-dimensional representations $\sigma_0,\dots,\sigma_n$32. Their main claim is that for affine Coxeter groups, the characteristic polynomial determined from an appropriate generating set determines the character of the representation (Feng et al., 16 May 2025). In that work, the affine Coxeter-element polynomial appears as part of a broader representation-theoretic program rather than as an invariant primarily tied to affine Lie algebras.

Within this broader context, the affine Coxeter polynomial of Damianou–Evripidou occupies a specific and classical role: it packages the eigenvalues of affine Coxeter transformations, connects directly to affine exponents and Coxeter numbers, admits explicit Chebyshev-polynomial derivations, and factors completely into cyclotomic polynomials (Damianou et al., 2014). The principal subtlety is not the existence of the invariant, but the coexistence of several neighboring conventions under similar names.