Reflection factorizations of Singer cycles

Lewis, Joel Brewster; Reiner, Victor; Stanton, Dennis

doi:10.46298/dmtcs.2401

Cited by 11 publications

(24 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Additional remarks on Theorem 1.4. The following remarks show how to recover the main theorems of [LRS14,HLR15] as special cases of Theorem 1.4. Incidentally, the first remark also settles a conjecture of Lewis-Reiner-Stanton.…”

Section: Factoring Regular Elliptic Elements Into Two Factorsmentioning

confidence: 99%

“…Recently, there has been interest in q-analogues of such problems, replacing S n with the finite general linear group GL n (F q ), the long cycle with a Singer cycle (or, more generally, regular elliptic element) c, and the number of cycles with the fixed space dimension [LRS14,HLR15]; or in more general geometric settings [HLRV11]. In the present paper, we extend this approach to give the following q-analogue of Theorem 1.1.…”

Section: Introductionmentioning

confidence: 99%

“…In GL n (F q ), the necessary character theory was worked out by Green [Gre55], building on work of Steinberg [Ste51]. This approach has been used recently by the first-named author and coauthors to count factorizations of Singer cycles into reflections [LRS14] and to count genus-0 factorizations (that is, those in which the codimensions of the fixed spaces of the factors sum to the codimension of the fixed space of the product) of regular elliptic elements [HLR15]. The current work subsumes these previous results while requiring no new character values.…”

Section: Introductionmentioning

confidence: 99%

“…) Such elements are called regular elliptic elements. They may be characterized in several other ways; see [LRS14,Prop. 4.4].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

GL_n(F_q)-analogues of factorization problems in the symmetric group

Lewis,

Morales

2015

Preprint

View full text Add to dashboard Cite

We consider GL n (F q )-analogues of certain factorization problems in the symmetric group S n : rather than counting factorizations of the long cycle (1, 2, . . . , n) given the number of cycles of each factor, we count factorizations of a regular elliptic element given the fixed space dimension of each factor. We show that, as in S n , the generating function counting these factorizations has attractive coefficients after an appropriate change of basis. Our work generalizes several recent results on factorizations in GL n (F q ) and also uses a character-based approach.As an application of our results, we compute the asymptotic growth rate of the number of factorizations of fixed genus of a regular elliptic element in GL n (F q ) into two factors as n → ∞. We end with a number of open questions.2. Regular elliptics, character theory, and the symmetric group approach 2.1. Singer cycles and regular elliptic elements. The field F q n is an n-dimensional vector space over F q , and multiplication by a fixed element in the larger field is a linear transformation.

show abstract

Section: Factoring Regular Elliptic Elements Into Two Factorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…) Such elements are called regular elliptic elements. They may be characterized in several other ways; see [LRS14,Prop. 4.4].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

GL_n(F_q)-analogues of factorization problems in the symmetric group

Lewis,

Morales

2015

Preprint

View full text Add to dashboard Cite

show abstract

“…Consider the regular elliptic elements of GL n F q , which are those matrices whose characteristic polynomial is irreducible over F q . Recent work by Huang, Lewis, Morales, Reiner, and Stanton [14,19,20] suggests that the regular elliptic elements are analogous to the n-cycles in S n from the perspective of enumerating factorizations.…”

Section: Introductionmentioning

confidence: 99%

Cycle type factorizations in $\mathrm{GL}_n \mathbb{F}_q$

Gordon

2020

Preprint

View full text Add to dashboard Cite

Recent work by Huang, Lewis, Morales, Reiner, and Stanton suggests that the regular elliptic elements of GL n F q are somehow analogous to the n-cycles of the symmetric group. In 1981, Stanley enumerated the factorizations of permutations into products of n-cycles. We study the analogous problem in GL n F q of enumerating factorizations into products of regular elliptic elements. More precisely, we define a notion of cycle type for GL n F q and seek to enumerate the tuples of a fixed number of regular elliptic elements whose product has a given cycle type. In some special cases, we provide explicit formulas, using a standard character-theoretic technique due to Frobenius by introducing simplified formulas for the necessary character values. We also address, for large q, the problem of computing the probability that the product of a random tuple of regular elliptic elements has a given cycle type. We conclude with some results about the polynomiality of our enumerative formulas and some open problems.

show abstract

Generating functions for weighted Hurwitz numbers

Guay-Paquet

Harnad

2017

Journal of Mathematical Physics

133

View full text Add to dashboard Cite

Double Hurwitz numbers enumerating weighted n-sheeted branched coverings of the Riemann sphere or, equivalently, weighted paths in the Cayley graph of S n generated by transpositions, are determined by an associated weight generating function. A uniquely determined 1-parameter family of 2D Toda τ -functions of hypergeometric type is shown to consist of generating functions for such weighted Hurwitz numbers. Four classical cases are detailed, in which the weighting is uniform: Okounkov's double Hurwitz numbers, for which the ramification is simple at all but two specified branch points; the case of Belyi curves, with three branch points, two with specified profiles; the general case, with a specified number of branch points, two with fixed profiles, the rest constrained only by the genus; and the signed enumeration case, with sign determined by the parity of the number of branch points. Using the exponentiated quantum dilogarithm function as weight generator, three new types of weighted enumerations are introduced. These determine quantum Hurwitz numbers depending on a deformation parameter q. By suitable interpretation of q, the statistical mechanics of quantum weighted branched covers may be related to that of Bosonic gases. The standard double Hurwitz numbers are recovered in the classical limit.

show abstract

Reflection factorizations of Singer cycles

Cited by 11 publications

References 0 publications

GL_n(F_q)-analogues of factorization problems in the symmetric group

GL_n(F_q)-analogues of factorization problems in the symmetric group

Cycle type factorizations in $\mathrm{GL}_n \mathbb{F}_q$

Generating functions for weighted Hurwitz numbers

Contact Info

Product

Resources

About