An official website of the United States government
Abstract A permutation statistic$${{\,\textrm{st}\,}}$$ is said to be shuffle-compatible if the distribution of$${{\,\textrm{st}\,}}$$ over the set of shuffles of two disjoint permutations$$\pi $$ and$$\sigma $$ depends only on$${{\,\textrm{st}\,}}\pi $$ ,$${{\,\textrm{st}\,}}\sigma $$ , and the lengths of$$\pi $$ and$$\sigma $$ . Shuffle-compatibility is implicit in Stanley’s early work onP-partitions, and was first explicitly studied by Gessel and Zhuang, who developed an algebraic framework for shuffle-compatibility centered around their notion of the shuffle algebra of a shuffle-compatible statistic. For a family of statistics called descent statistics, these shuffle algebras are isomorphic to quotients of the algebra of quasisymmetric functions. Recently, Domagalski, Liang, Minnich, Sagan, Schmidt, and Sietsema defined a version of shuffle-compatibility for statistics on cyclic permutations, and studied cyclic shuffle-compatibility through purely combinatorial means. In this paper, we define the cyclic shuffle algebra of a cyclic shuffle-compatible statistic, and develop an algebraic framework for cyclic shuffle-compatibility in which the role of quasisymmetric functions is replaced by the cyclic quasisymmetric functions recently introduced by Adin, Gessel, Reiner, and Roichman. We use our theory to provide explicit descriptions for the cyclic shuffle algebras of various cyclic permutation statistics, which in turn gives algebraic proofs for their cyclic shuffle-compatibility.
more »
« less
On kernels of descent statistics
The kernel $$\mathcal{K}^{\operatorname{st}}$$ of a descent statistic $$\operatorname{st}$$, introduced by Grinberg, is a subspace of the algebra $$\operatorname{QSym}$$ of quasisymmetric functions defined in terms of $$\operatorname{st}$$-equivalent compositions, and is an ideal of $$\operatorname{QSym}$$ if and only if $$\operatorname{st}$$ is shuffle-compatible. This paper continues the study of kernels of descent statistics, with emphasis on the peak set $$\operatorname{Pk}$$ and the peak number $$\operatorname{pk}$$. The kernel $$\mathcal{K}^{\operatorname{Pk}}$$ in particular is precisely the kernel of the canonical projection from $$\operatorname{QSym}$$ to Stembridge's algebra of peak quasisymmetric functions, and is the orthogonal complement of Nyman's peak algebra. We prove necessary and sufficient conditions for obtaining spanning sets and linear bases for the kernel $$\mathcal{K}^{\operatorname{st}}$$ of any descent statistic $$\operatorname{st}$$ in terms of fundamental quasisymmetric functions, and give characterizations of $$\mathcal{K}^{\operatorname{Pk}}$$ and $$\mathcal{K}^{\operatorname{pk}}$$ in terms of the fundamental basis and the monomial basis of $$\operatorname{QSym}$$. Our results imply that the peak set and peak number statistics are $$M$$-binomial, confirming a conjecture of Grinberg.
more »
« less
- Award ID(s):
- 2316181
- PAR ID:
- 10516023
- Publisher / Repository:
- Electronic Journal of Combinatorics
- Date Published:
- Journal Name:
- Electronic Journal of Combinatorics
- Volume:
- 31
- Issue:
- 2
- ISSN:
- 1077-8926
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract Given a permutation statistic$$\operatorname {\mathrm {st}}$$, define its inverse statistic$$\operatorname {\mathrm {ist}}$$by. We give a general approach, based on the theory of symmetric functions, for finding the joint distribution of$$\operatorname {\mathrm {st}}_{1}$$and$$\operatorname {\mathrm {ist}}_{2}$$whenever$$\operatorname {\mathrm {st}}_{1}$$and$$\operatorname {\mathrm {st}}_{2}$$are descent statistics: permutation statistics that depend only on the descent composition. We apply this method to a number of descent statistics, including the descent number, the peak number, the left peak number, the number of up-down runs and the major index. Perhaps surprisingly, in many cases the polynomial giving the joint distribution of$$\operatorname {\mathrm {st}}_{1}$$and$$\operatorname {\mathrm {ist}}_{2}$$can be expressed as a simple sum involving products of the polynomials giving the (individual) distributions of$$\operatorname {\mathrm {st}}_{1}$$and$$\operatorname {\mathrm {st}}_{2}$$. Our work leads to a rederivation of Stanley’s generating function for doubly alternating permutations, as well as several conjectures concerning real-rootedness and$$\gamma $$-positivity.more » « less
-
We introduce a new basis of quasisymmetric functions, the row-strict dual immaculate functions. We construct a cyclic, indecomposable 0-Hecke algebra module for these functions. Our row-strict immaculate functions are related to the dual immaculate functions of Berg-Bergeron-Saliola-Serrano-Zabrocki (2014-15) by the involution on the ring of quasisymmetric functions. We give an explicit description of the effect of on the associated 0-Hecke modules, via the poset induced by the 0-Hecke action on standard immaculate tableaux. This remarkable poset reveals other 0-Hecke submodules and quotient modules, often cyclic and indecomposable, notably for a row-strict analogue of the extended Schur functions studied in Assaf-Searles (2019). Like the dual immaculate function, the row-strict dual immaculate function is the generating function of a suitable set of tableaux, corresponding to a specific descent set. We give a complete combinatorial and representation-theoretic picture by constructing 0-Hecke modules for the remaining variations on descent sets, and showing thatallthe possible variations for generating functions of tableaux occur as characteristics of the 0-Hecke modules determined by these descent sets.more » « less
-
Abstract We prove an inequality between the conductor and the discriminant for all hyperelliptic curves defined over discretely valued fields $$K$$ with perfect residue field of characteristic not $$2$$. Specifically, if such a curve is given by $$y^{2} = f(x)$$ with $$f(x) \in \mathcal{O}_{K}[x]$$, and if $$\mathcal{X}$$ is its minimal regular model over $$\mathcal{O}_{K}$$, then the negative of the Artin conductor of $$\mathcal{X}$$ (and thus also the number of irreducible components of the special fiber of $$\mathcal{X}$$) is bounded above by the valuation of $$\operatorname{disc}(f)$$. There are no restrictions on genus of the curve or on the ramification of the splitting field of $$f$$. This generalizes earlier work of Ogg, Saito, Liu, and the second author.more » « less
-
Abstract The well-studied moduli space of complex cubic surfaces has three different, but isomorphic, compact realizations: as a GIT quotient$${\mathcal {M}}^{\operatorname {GIT}}$$, as a Baily–Borel compactification of a ball quotient$${(\mathcal {B}_4/\Gamma )^*}$$, and as a compactifiedK-moduli space. From all three perspectives, there is a unique boundary point corresponding to non-stable surfaces. From the GIT point of view, to deal with this point, it is natural to consider the Kirwan blowup$${\mathcal {M}}^{\operatorname {K}}\rightarrow {\mathcal {M}}^{\operatorname {GIT}}$$, whereas from the ball quotient point of view, it is natural to consider the toroidal compactification$${\overline {\mathcal {B}_4/\Gamma }}\rightarrow {(\mathcal {B}_4/\Gamma )^*}$$. The spaces$${\mathcal {M}}^{\operatorname {K}}$$and$${\overline {\mathcal {B}_4/\Gamma }}$$have the same cohomology, and it is therefore natural to ask whether they are isomorphic. Here, we show that this is in factnotthe case. Indeed, we show the more refined statement that$${\mathcal {M}}^{\operatorname {K}}$$and$${\overline {\mathcal {B}_4/\Gamma }}$$are equivalent in the Grothendieck ring, but notK-equivalent. Along the way, we establish a number of results and techniques for dealing with singularities and canonical classes of Kirwan blowups and toroidal compactifications of ball quotients.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.37236/12164
