The question of whether polynomial time algorithms exist for finding double coset representatives in symmetric groups sits at a fascinating crossroads of algebraic structure and computational complexity. Double cosets—sets of the form HxK where H and K are subgroups of a group G, and x is an element of G—play a pivotal role in areas ranging from representation theory to combinatorics and even the structure of algebraic number fields. But when it comes to the symmetric group S_n, the heart of permutation group theory, the computational status of double coset representatives is subtle and shaped by both the richness of their algebraic structure and the hardness of underlying algorithmic problems.
Short answer: In general, there are **no known polynomial time algorithms** for finding representatives of all double cosets H\G/K in the symmetric group S_n for arbitrary subgroups H and K. The problem is believed to be computationally hard in the general case, potentially NP-complete, as suggested by classic results. However, for certain specific cases—most notably when H and K are so-called Young subgroups (parabolic subgroups), or are centralizers of particular elements—efficient, often explicit constructions for double coset representatives are possible, sometimes even polynomial in n. The feasibility thus hinges crucially on the structure of H and K.
Let’s unravel why this is the case, exploring both the computational obstacles and the special cases where progress is possible.
Understanding Double Cosets in Symmetric Groups
Double cosets in the symmetric group S_n arise naturally when studying how two subgroups, H and K, interact within the group of all permutations of n elements. A double coset HxK is the set of all products h x k where h is in H and k is in K, and x is a fixed element of S_n. The set of all such double cosets, denoted H\S_n/K, partitions the group S_n into equivalence classes. These partitions are central in representation theory (as seen in Mackey’s decomposition theorem, discussed at ncatlab.org), combinatorics, and the study of induced representations.
The general problem is: given subgroups H, K ≤ S_n, can you efficiently (that is, in time polynomial in n) compute a set of representatives {x_i} such that every element of S_n lies in exactly one double coset Hx_iK?
Computational Hardness: The General Case
According to cs.stackexchange.com, the general computational problem of finding double coset representatives in S_n is suspected to be hard. The referenced work of Luks, a foundational figure in permutation group algorithms, suggests that the problem is "NP-complete" in the worst case. This implies that, unless P equals NP (a central open question in computer science), no polynomial time algorithm is likely to exist for all possible choices of H and K. The hardness stems from the sheer size and complexity of the symmetric group—S_n has n! elements, and the number and structure of its subgroups grow explosively with n.
This computational hardness is not just theoretical. In practice, as highlighted in mathoverflow.net’s discussion on using computational algebra systems like Magma, even for moderately sized groups, algorithms to enumerate double coset representatives can be slow and may rely on randomization or brute force enumeration. Moreover, as Derek Holt points out, subtle issues in subgroup conjugacy and action conventions can further complicate matters, making fully deterministic, efficient algorithms elusive for general subgroups.
Polynomial Time for Special Subgroups: Young Subgroups and Parabolic Cases
Yet, the story brightens considerably for important classes of subgroups, particularly the so-called Young subgroups of S_n. These are subgroups that fix certain blocks of elements and are key objects in the combinatorics of symmetric groups and the representation theory of GL_n. As explained on mathoverflow.net, for pairs of Young subgroups S_λ and S_μ (associated to compositions λ and μ of n), there is a canonical, efficient way to describe double coset representatives.
Specifically, S_n is a Coxeter group (generated by adjacent transpositions), and Young subgroups are its parabolic subgroups. In this setting, there is always a unique element of minimal length (in the sense of the Coxeter group’s generating set) in each double coset. This minimal representative can be constructed algorithmically, often in time polynomial in n. Johannes Hahn notes that this approach "provides algorithmic ways to efficiently iterate through the set of (double)cosets by enumerating the representatives." To make this concrete, for compositions λ = (λ_1, λ_2) and μ = (μ_1, μ_2), the number of double cosets S_λ\S_n/S_μ is simply min{λ_1, λ_2, μ_1, μ_2} + 1, and representatives can be written down explicitly using matrix or combinatorial descriptions.
Moreover, as Zhibin elaborates, this explicit construction is tied to the so-called Bruhat decomposition and can be described in terms of inequalities among the entries of permutations—conditions like w(i) < w(j) for certain positions—making the representatives not just efficiently computable but also combinatorially transparent.
Centralizers and Other Special Subgroups
Another case of interest, mentioned in cs.stackexchange.com, is when H and K are centralizers of elements in S_n. Centralizers often have special structure, which can sometimes be exploited for efficient computation. However, the existence of polynomial time algorithms in these cases depends on the specific elements being centralized and the resulting subgroup structure. In some scenarios, the problem reduces to the parabolic (Young subgroup) case; in others, it remains as hard as the general problem. No general polynomial time algorithm is known for all centralizer pairs, but for many “nice” cases, efficient enumeration is possible.
Theoretical and Practical Significance
The importance of double coset representatives goes far beyond mere group partitioning. As ncatlab.org and math.stackexchange.com both note, double cosets underpin major results in representation theory, like Mackey’s formula, which describes how to restrict induced representations from one subgroup to another. In combinatorics and algebraic number theory, they arise in the analysis of Burnside rings, the construction of Hecke algebras, and even in describing how prime ideals decompose in Galois extensions (as described by W Sao on mathoverflow.net).
Because of these deep connections, efficient algorithms for double coset representatives—where they exist—have practical computational consequences for fields as diverse as computational group theory, algebraic combinatorics, and computational number theory.
Contrast: The Role of Computational Algebra Systems
In practice, computational algebra systems like Magma and GAP provide built-in routines for double coset computations. However, as discussed on mathoverflow.net, these algorithms may use randomization and can yield different outputs on different runs unless the random seed is fixed. The underlying methods are typically exponential in the worst case, but may be practical for small groups or groups with “nice” subgroup structure.
A telling example is the calculation for the group M10 (a permutation group of order 720), where finding double coset representatives for subgroups of order 72 and 2 was feasible in Magma for small cases, but became unreliable or inconsistent for larger or more complex subgroups without careful attention to group action conventions and code correctness.
Summary Table: When Are Polynomial Algorithms Known?
To summarize the landscape:
- For arbitrary subgroups H, K ≤ S_n: No known polynomial time algorithm; the problem is suspected to be hard (potentially NP-complete). - For Young subgroups (parabolic case): Explicit, efficient constructions are known; minimal length representatives can be found in polynomial time. - For centralizers: No general polynomial time algorithm, but special cases may reduce to the parabolic case. - For specific small groups or highly structured subgroups: Computation is feasible in practice, though not always polynomial in the worst case.
“Each double coset has exactly one such representative” (mathoverflow.net), and for Young subgroups, “there is a canonical way to describe coset and double-coset representatives” (mathoverflow.net). However, “the problem is NP-complete” in the general case (cs.stackexchange.com), and “algorithms... make use of random choices of group elements” (mathoverflow.net), reflecting the practical difficulties.
Closing Thoughts
The search for efficient algorithms for double coset representatives touches on deep mathematical currents—algebraic structure, combinatorial optimization, and computational complexity. While the symmetric group’s rich structure allows for beautiful explicit answers in some cases, the shadow of computational hardness looms large in the general case. If your application involves Young subgroups or similarly structured subgroups, you are in luck: polynomial time algorithms exist, and the combinatorics is well understood. For arbitrary subgroups, the best that current mathematics and computer science can offer is exponential time enumeration, with the tantalizing possibility that some future breakthrough might change our understanding of the computational landscape.
For now, the answer is clear: **polynomial time algorithms exist for finding double coset representatives in symmetric groups only in special cases—most notably for Young subgroups—while the general problem remains computationally intractable.** This status is backed by both theoretical results and practical experience, as seen across sources like cs.stackexchange.com, mathoverflow.net, and ncatlab.org.