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Finite Gröbner bases for quantum symmetric groups

Leonard Schmitz, Marcel Wack

TL;DR

The paper proves that the two-sided ideal $I_n$ defining Wang's algebraic quantum symmetric group admits a finite, monic, reduced Gröbner basis $G_n$ for all $n\ge 4$, with cardinality $4n^3-15n^2+16n-2$, enabling a decidable word problem in $\mathfrak S_n$. The authors develop a constructive proof via interreduction to a compact generating set, introduce orthogonal refinements, and assemble a full Gröbner basis from a controlled set of overlaps, verifying all cases through Buchberger-type criteria and Gröbner certificates. To manage the large, parameterized computations, they implement a general computational framework in OSCAR that reduces the problem to a $\mathbb{Z}$-module with logically independent predicates, and certify key reductions (notably $\mathsf{rwel}_{23}$) using predicate-based preimages. The result is a scalable, closed-form basis for an entire family of non-trivial ideals, with potential implications for quantum symmetry problems in matroids and related combinatorial structures, and it opens avenues for efficient decision procedures in non-commutative quotient algebras.

Abstract

Non-commutative Gröbner bases of two-sided ideals are not necessarily finite. Motivated by this, we provide a closed-form description of a finite and reduced Gröbner bases for the two-sided ideal used in the construction of Wangs quantum symmetric group. In particular, this proves that the word problem for quantum symmetric groups is decidable.

Finite Gröbner bases for quantum symmetric groups

TL;DR

The paper proves that the two-sided ideal defining Wang's algebraic quantum symmetric group admits a finite, monic, reduced Gröbner basis for all , with cardinality , enabling a decidable word problem in . The authors develop a constructive proof via interreduction to a compact generating set, introduce orthogonal refinements, and assemble a full Gröbner basis from a controlled set of overlaps, verifying all cases through Buchberger-type criteria and Gröbner certificates. To manage the large, parameterized computations, they implement a general computational framework in OSCAR that reduces the problem to a -module with logically independent predicates, and certify key reductions (notably ) using predicate-based preimages. The result is a scalable, closed-form basis for an entire family of non-trivial ideals, with potential implications for quantum symmetry problems in matroids and related combinatorial structures, and it opens avenues for efficient decision procedures in non-commutative quotient algebras.

Abstract

Non-commutative Gröbner bases of two-sided ideals are not necessarily finite. Motivated by this, we provide a closed-form description of a finite and reduced Gröbner bases for the two-sided ideal used in the construction of Wangs quantum symmetric group. In particular, this proves that the word problem for quantum symmetric groups is decidable.

Paper Structure

This paper contains 12 sections, 35 theorems, 91 equations, 4 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. The main result
  3. Quantum symmetric groups
  4. Primer on non-commutative Gröbner bases
  5. Proof of Theorem \ref{['thm:finite_gb']}
  6. An interreduced generating set for In
  7. Construction of the Gröbner basis TEXT
  8. Remaining overlap relations in TEXT
  9. A general computational proof using OSCAR
  10. Finitely generated modules using predicates
  11. Reduction of rwel23 modulo Fn
  12. Outlook and future work

Key Result

Theorem 1.2

For all $n\geq 4$ the ideal $I_n$ has a finite, monic and reduced Gröbner basis with respect to the graded lexicographic order via row-wise ordering in $(u_{ij})_{1\leq i,j\leq n}$. Its explicit cardinality is given by the cubic polynomial $\# G_n=4n^3 -15n^2 +16n-2$.

Figures (4)

  • Figure 1: List of all possible overlaps between families in $F_n$.
  • Figure 2: The two possible overlaps of the paring $\mathsf{inj}$ and $\mathsf{rinj}$, resulting in the two families of overlap relations, $\mathsf{bg}^{(1)}$ and $\mathsf{bg}^{(2)}$.
  • Figure 3: Graph of all possible overlap parings between families in $F_n$.
  • Figure 4: Graph of all possible overlap parings between families in $G_n$ that have not been covered by \ref{['fig:pentagonGraphG0']}. The edges marked with $\times$ follow by transposition.

Theorems & Definitions (75)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 65 more