Lee and Lee (2019) constructed by hand a finite Gröbner–Shirshov basis for the group algebra of the finite non-crystallographic Coxeter group H4H_4 (with H3H_3 and its parabolic H2H_2 treated along the way), enumerated the 14400 standard monomials, and sketched — but did not print — the resulting multiplication table. This problem asks for the natural extension of that program: construct Gröbner–Shirshov bases and standard-monomial normal forms for the affine non-crystallographic Coxeter groups H~2\widetilde{H}_2, H~3\widetilde{H}_3, H~4\widetilde{H}_4, which are infinite and so require parametrized families of rewriting rules. The affine extension is the target: the complete multiplication table of the 14400 standard monomials of H4H_4 is a routine finite computation determined by the paper's own rewriting rules (the paper itself states that the table 'follows naturally'), and general-purpose completion algorithms exist for the finite types.

Background

A Coxeter group is a group generated by involutions s1,…,sns_1,\dots,s_n subject only to braid relations of the form (sisj)mij=1(s_i s_j)^{m_{ij}}=1, where the Coxeter matrix (mij)(m_{ij}) encodes the angles between reflecting hyperplanes. The finite Coxeter groups were classified by Coxeter in 1935. Most of them are crystallographic (Weyl groups), meaning they stabilize a lattice; the three non-crystallographic finite types are H2H_2 (the dihedral group of order 10, symmetries of a regular pentagon), H3H_3 (the full icosahedral group of order 120), and H4H_4 (the symmetry group of the 120-cell/600-cell pair, of order 14400). Their failure to be crystallographic is tied to the golden ratio in the m=5m=5 bond, and their root systems underlie quasicrystals with five-fold diffraction symmetry. The affine Coxeter groups are the infinite cousins obtained by adding one reflection to a finite diagram; in the crystallographic case (the affine Weyl groups) each decomposes as a semidirect product of a lattice-translation group by the finite Weyl group it extends. The affine extensions H~k\widetilde{H}_k of the non-crystallographic types were constructed concretely by Patera and Twarock (2002), via extended Cartan matrices, as reflection groups of infinite order that are used as symmetry groups in mathematical models for quasicrystal fragments; unlike the crystallographic affine case, the orbit sets they generate are not lattices but cover the whole space densely, and the construction of finite quasicrystal fragments in that work is explicitly contrasted with the cut-and-project scheme. Their rewriting-theoretic normal forms have not been worked out.

A Gröbner–Shirshov basis is the noncommutative analogue of a Gröbner basis: for an algebra presented as a quotient F⟨X⟩/IF\langle X\rangle/I of the free associative algebra by relations, it is a set SS of monic polynomials generating II whose leading monomials are pairwise composable in such a way that every overlap ambiguity (composition) reduces to zero; by the Composition Lemma (Shirshov; Bergman's diamond lemma), the SS-standard monomials — the words irreducible under the induced rewriting rules — then form a vector-space basis of the algebra, giving canonical normal forms. For a finite group GG, a Gröbner–Shirshov basis of the group algebra C[G]\mathbb{C}[G] can be certified by counting: if the irreducible words number exactly ∣G∣|G|, the basis is complete. Bokut and Shiao (2001) and subsequent authors (Lee for E6,E7,F4E_6,E_7,F_4; Pang and Wang for G2,F4,E6,E7,E8G_2,F_4,E_6,E_7,E_8) built such bases for the crystallographic finite Coxeter groups, and the affine Weyl types A~n\widetilde{A}_n (Yilmaz–Özel–Ustaoğlu, 2014) and C~n\widetilde{C}_n (Ustaoğlu–Yılmaz, 2022) were handled with infinite, parametrized rule families.

Lee and Lee (2019) did the non-crystallographic finite case: they completed the Coxeter presentation of H3H_3 (via Lemma 3.2) and H4H_4 (via Lemmas 4.1–4.16) to an explicit Gröbner–Shirshov basis under a degree-lexicographic order (words compared first by length, then alphabetically), enumerated the 10⋅12⋅120=1440010\cdot 12\cdot 120=14400 standard monomials of H4H_4, identified the coset representatives of the chains H2⊂H3⊂H4H_2\subset H_3\subset H_4, and observed that the multiplication table of the standard monomials 'follows naturally' — without printing it. Their proofs are hand derivations with case-by-case composition checks. Later work by the same school extended the technique to the rank-three non-Coxeter Shephard groups (complex reflection analogues), producing monomial bases with explicit multiplication tables for those much smaller groups (Kim–Lee, 2022), but left the affine extension above untouched. Note that general algorithms give weaker, non-explicit information: Brink and Howlett (1993) proved every finitely generated Coxeter group — including the H~k\widetilde{H}_k — is automatic, meaning its word-reduction behavior is governed by finite-state automata so that algorithmic normal forms exist in principle, but no explicit Gröbner–Shirshov basis or standard-monomial description for the affine non-crystallographic types is available.

Problem Statement

Complete the Gröbner–Shirshov normal-form program that Lee–Lee (2019) began for the non-crystallographic Coxeter groups by extending it to the affine types. Concretely: for each of the affine Coxeter groups H~2\widetilde{H}_2, H~3\widetilde{H}_3, H~4\widetilde{H}_4 (the rank-(k+1)(k{+}1) affine extensions of HkH_k, as constructed by Patera–Twarock), fix an explicit monomial order on the free associative algebra of the standard Coxeter generators, construct a Gröbner–Shirshov basis of the group algebra — since these groups are infinite, the basis is expected to be a finite or finitely parametrized family of rewriting rules — and characterize the standard monomials (the irreducible words giving canonical normal forms).

This problem concerns the affine groups: the Lee–Lee bases for H3H_3 and H4H_4 and the enumeration of the 10⋅12⋅120=1440010\cdot 12\cdot 120=14400 standard monomials of H4H_4 are established background and are not part of the target.

References

  1. Jeong-Yup Lee, Dong-il Lee, Gröbner–Shirshov bases for non-crystallographic Coxeter groups, Acta Crystallographica Section A: Foundations and Advances 75(3) (2019) 584–592, DOI 10.1107/S2053273319002092, https://doi.org/10.1107/S2053273319002092 (preprint arXiv:1809.08419, https://arxiv.org/abs/1809.08419)
  2. L. A. Bokut, L.-S. Shiao, Gröbner–Shirshov bases for Coxeter groups, Communications in Algebra 29(9) (2001) 4305–4319, DOI 10.1081/AGB-100106002, https://doi.org/10.1081/AGB-100106002
  3. Sungsoon Kim, Dong-il Lee, Monomial bases for the primitive complex Shephard groups of rank three, Communications in Algebra 50(2) (2022) 889–902, DOI 10.1080/00927872.2021.1976200, https://doi.org/10.1080/00927872.2021.1976200
  4. Dong-Il Lee, A Computational Approach to Shephard Groups, in: Rings, Monoids and Module Theory, Springer Proceedings in Mathematics & Statistics 382, Springer (2021) 287–305, DOI 10.1007/978-981-16-8422-7_16, https://doi.org/10.1007/978-981-16-8422-7_16
  5. Xiaowei Pang, Jun Wang, Gröbner-Shirshov bases for the Coxeter groups of the types G2,F4,E6,E7G_2, F_4, E_6, E_7 and E8E_8, arXiv:2002.07369 (2020, revised 2026), https://arxiv.org/abs/2002.07369
  6. Erol Yilmaz, Cenap Özel, Uğur Ustaoğlu, Gröbner–Shirshov basis and reduced words for affine Weyl group A~n\widetilde{A}_n, Journal of Algebra and Its Applications 13(6) (2014) 1450005, DOI 10.1142/S0219498814500054, https://doi.org/10.1142/S0219498814500054
  7. Uğur Ustaoğlu, Erol Yılmaz, Gröbner–Shirshov basis and normal forms for the infinite Coxeter group of type C~n\widetilde{C}_n, Advanced Studies: Euro-Tbilisi Mathematical Journal 15(2) (2022) 89–99, DOI 10.32513/asetmj/19322008218, https://doi.org/10.32513/asetmj/19322008218
  8. Uğur Ustaoğlu, Erol Yılmaz, Gröbner–Shirshov bases and normal forms of some Coxeter groups, JP Journal of Algebra, Number Theory and Applications 40(5) (2018) 701–723, DOI 10.17654/nt040050701, https://doi.org/10.17654/nt040050701
  9. Jiří Patera, Reidun Twarock, Affine extension of noncrystallographic Coxeter groups and quasicrystals, Journal of Physics A: Mathematical and General 35(7) (2002) 1551–1574, DOI 10.1088/0305-4470/35/7/306, https://doi.org/10.1088/0305-4470/35/7/306
  10. Brigitte Brink, Robert B. Howlett, A finiteness property and an automatic structure for Coxeter groups, Mathematische Annalen 296 (1993) 179–190, DOI 10.1007/BF01445101, https://doi.org/10.1007/BF01445101