Building on work of Müger and Tuset [arXiv.2210.06582 (2022)], we reduce the Mathieu conjecture, formulated by Mathieu in 1997 [Algebre Non Commutative, Groupes Quantiques et Invariants, edited by Alex, J. and Cauchon, G. (Société Mathématique de France, Reims, 1997), Vol. 2, pp. 263–279], for SU(N) to a simpler conjecture in purely abelian terms. We sketch a similar reduction for SO(N). The proofs rely on Euler-style parametrizations of these groups, which we discuss including proofs.
I. INTRODUCTION
In a 1997 paper, Mathieu conjectured the following statement:
(Ref. 8). Let G be a compact connected Lie group. If f, h are finite-type functions such that ∫GfPdg = 0 for all , then ∫GfPhdg = 0 for all large enough P.
In 1998, only one year after the publication of Mathieu’s paper, Duistermaat and van der Kallen4 proved Mathieu’s conjecture for all abelian connected compact groups. While the conjecture still has not been proven for any non-abelian group, some attempts were made. Dings and Koelink3 approached the conjecture for SU(2) by expressing the finite-type functions by explicit matrix coefficients. Influenced heavily by this, Müger and Tuset9 reduced the Mathieu conjecture for SU(2) to a conjecture about certain Laurent polynomials.
The goal of the present paper is to generalize the paper by Müger and Tuset to the compact matrix groups SU(N) and SO(N), where N ≥ 2. A key ingredient to achieving this will be a generalization of the Euler decomposition. The Euler decomposition for SU(2) has been known for some time but is mostly used by physicists under the name of Euler angles. This is no different in the case of SU(N). Several (similar but not equal) versions of the Euler decomposition for SU(N) exist, see for example Bertini et al.,1 Cacciatori et al.,2 or Tilma and Sudarshan.13 In a similar way there exist several decompositions of SO(N), see for example Refs. 6, 10, and 14.
In our paper, we will reduce the Mathieu conjecture to a conjecture similar to that of Müger and Tuset.9 We start by looking at the matrix coefficients of the generalized Euler decomposition on SU(N) and SO(N), and we find that any finite-type function can be described by a function on . To be more specific, any finite type function reduces to a function which can be written as where is a multi-index where for each i, and is a polynomial in x1, …, xk and . Assuming these functions satisfy other conjectures, the Mathieu conjecture is proven for SU(N) and SO(N). The proof uses the explicit description of the Euler decomposition on SU(N) and SO(N) and the properties of the Haar measure in these parametrizations. In Sec. II we will focus on the group SU(N), while in Sec. III the group SO(N) will be considered. The final part of the paper is dedicated to proving the generalized Euler decomposition we used throughout this paper, with the corresponding explicit description of the Haar measure in this parametrization.
II. THE CASE OF SU(N)
In this paper we will reduce the Mathieu’s conjecture on SU(N) and SO(N) with N ≥ 2. We start by recalling Mathieu’s conjecture. To do so, we first introduce the notion of a finite-type function:
(The Mathieu conjecture8). Let G be a compact connected Lie group. If f, h are finite-type functions such that ∫GfP dg = 0 for all , then ∫GfPh dg = 0 for all large enough P.
In this section we will focus on SU(N). We will base our parametrization and Haar measure on Refs. 12 and 13. For completeness, we included in the Appendix dedicated to proving the parametrization.
We note that if we restrict any finite type function h to a closed subgroup H of SU(N), then h|H also is a finite-type function. This can easily be seen by the fact that any irreducible representation (π, V) of SU(N) is finite dimensional, hence (π|H, V) splits into finitely many irreducible representations (πH,i, Vi) of H, i.e., . It is immediate then that h|H is again a finite-type function.
The main ingredients of the Proof of Lemma 2.7 are captured in the following lemma:
Both equalities can be found by using a subsitution. The former integral is found by setting z = eiϕ and the latter by x = sin(ϕ).□
In other words, we have translated the problem of the non-abelian group SU(N) to the simpler set . This is used to translate Mathieu’s conjecture to a complex analysis question in the case of SU(N).
It is clear that is a SU(N)-admissible function, so we focus on this class of functions. Motivated by Ref. 9, we make the following conjecture:
At first sight, this conjecture may seem to have little to do with Mathieu’s conjecture. However
Assume Conjecture 2.10 is true. Then Mathieu’s conjecture is true for SU(N).
One could wonder whether the inverse implication of Theorem 2.11 is true as well. It is still an open question whether this is the case.
III. THE CASE OF SO(N)
The SO(N) finite-type functions differ slightly from the SU(N) finite-type functions in Eq. (2.3). We chose to have (possible) higher powers of cos(ϕj) instead of sin(ϕj) because the Haar measure only contains powers of sin(ϕj). In addition, there are fewer parameters going over [0, 2π], hence we can only write ϕ1, ϕN, …, ϕN(N−1)/2 as an exponential.
In the same way as with SU(N), we can translate the problem then back to analysis of functions on in the following way:
Note that Lemma 3.3 is similar to Lemma 2.7, the difference here being that the xj variables go over the interval [−1, 1] instead of [0,1] which is due to the original intervals being [0, π] instead of [0, π/2] in the SU(N) case. The proof goes identical to the Proof of Lemma 2.7.
Similar to Sec. II, we describe a conjecture that will deal with Mathieu’s conjecture for SO(N):
Assume Conjecture 3.5 is true. Then Mathieu’s conjecture is true for SO(N).
ACKNOWLEDGMENTS
The author would like to thank Michael Müger for proposing the subject, and the many valuable discussions we had. He also wishes to thank Erik Koelink for feedback and suggestions.
AUTHOR DECLARATIONS
Conflict of Interest
The author has no conflicts to disclose.
Author Contributions
Kevin Zwart: Conceptualization (equal); Writing – original draft (equal); Writing – review & editing (equal).
DATA AVAILABILITY
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
APPENDIX: PROOF OF LEMMA 2.3 AND LEMMA 2.5
In this appendix, we will prove Lemma 2.3 and Lemma 2.5. We will note that the proofs are inspired by and/or based on Refs. 1 and 15–13.
Second, we construct left-invariant one-forms on G/K, which can be wedged to find dgK. Third, we will show how the top form dgK looks like explicitly by considering the parameterization of SU(N) as in Lemma 2.3. We end the proof by normalizing the measure to get the Haar measure, which we shall call dgSU(N).