60 research outputs found
Strong forms of self-duality for Hopf monoids in species
A vector species is a functor from the category of finite sets with
bijections to vector spaces (over a fixed field); informally, one can view this
as a sequence of -modules. A Hopf monoid (in the category of vector
species) consists of a vector species with unit, counit, product, and coproduct
morphisms satisfying several compatibility conditions, analogous to a graded
Hopf algebra. A vector species has a basis if and only if it is given by a
sequence of -modules which are permutation representations. We say that a
Hopf monoid is freely self-dual if it is connected and finite-dimensional, and
if it has a basis in which the structure constants of its product and coproduct
coincide. Such Hopf monoids are self-dual in the usual sense, and we show that
they are furthermore both commutative and cocommutative. We prove more specific
classification theorems for freely self-dual Hopf monoids whose products
(respectively, coproducts) are linearized in the sense that they preserve the
basis; we call such Hopf monoids strongly self-dual (respectively, linearly
self-dual). In particular, we show that every strongly self-dual Hopf monoid
has a basis isomorphic to some species of block-labeled set partitions, on
which the product acts as the disjoint union. In turn, every linearly self-dual
Hopf monoid has a basis isomorphic to the species of maps to a fixed set, on
which the coproduct acts as restriction. It follows that every linearly
self-dual Hopf monoid is strongly self-dual. Our final results concern
connected Hopf monoids which are finite-dimensional, commutative, and
cocommutative. We prove that such a Hopf monoid has a basis in which its
product and coproduct are both linearized if and only if it is strongly
self-dual with respect to a basis equipped with a certain partial order,
generalizing the refinement partial order on set partitions.Comment: 42 pages; v2: a few typographical errors corrected and references
updated; v3: discussion in Sections 3.1 and 3.2 slightly revised, Theorem A
corrected to include hypothesis about ambient field, final versio
Crossings and nestings in colored set partitions
Chen, Deng, Du, Stanley, and Yan introduced the notion of -crossings and
-nestings for set partitions, and proved that the sizes of the largest
-crossings and -nestings in the partitions of an -set possess a
symmetric joint distribution. This work considers a generalization of these
results to set partitions whose arcs are labeled by an -element set (which
we call \emph{-colored set partitions}). In this context, a -crossing or
-nesting is a sequence of arcs, all with the same color, which form a
-crossing or -nesting in the usual sense. After showing that the sizes of
the largest crossings and nestings in colored set partitions likewise have a
symmetric joint distribution, we consider several related enumeration problems.
We prove that -colored set partitions with no crossing arcs of the same
color are in bijection with certain paths in \NN^r, generalizing the
correspondence between noncrossing (uncolored) set partitions and 2-Motzkin
paths. Combining this with recent work of Bousquet-M\'elou and Mishna affords a
proof that the sequence counting noncrossing 2-colored set partitions is
P-recursive. We also discuss how our methods extend to several variations of
colored set partitions with analogous notions of crossings and nestings.Comment: 25 pages; v2: material revised and condensed; v3 material further
revised, additional section adde
A symplectic refinement of shifted Hecke insertion
Buch, Kresch, Shimozono, Tamvakis, and Yong defined Hecke insertion to
formulate a combinatorial rule for the expansion of the stable Grothendieck
polynomials indexed by permutations in the basis of stable Grothendieck
polynomials indexed by partitions. Patrias and Pylyavskyy
introduced a shifted analogue of Hecke insertion whose natural domain is the
set of maximal chains in a weak order on orbit closures of the orthogonal group
acting on the complete flag variety. We construct a generalization of shifted
Hecke insertion for maximal chains in an analogous weak order on orbit closures
of the symplectic group. As an application, we identify a combinatorial rule
for the expansion of "orthogonal" and "symplectic" shifted analogues of
in Ikeda and Naruse's basis of -theoretic Schur -functions.Comment: 40 pages; v2: fixed several errors, minor reorganization; v3: further
corrections, condensed expositio
Combinatorial methods of character enumeration for the unitriangular group
Let \UT_n(q) denote the group of unipotent upper triangular
matrices over a field with elements. The degrees of the complex irreducible
characters of \UT_n(q) are precisely the integers with , and it has been
conjectured that the number of irreducible characters of \UT_n(q) with degree
is a polynomial in with nonnegative integer coefficients (depending
on and ). We confirm this conjecture when and is arbitrary
by a computer calculation. In particular, we describe an algorithm which allows
us to derive explicit bivariate polynomials in and giving the number of
irreducible characters of \UT_n(q) with degree when and . When divided by and written in terms of the variables
and , these functions are actually bivariate polynomials with nonnegative
integer coefficients, suggesting an even stronger conjecture concerning such
character counts. As an application of these calculations, we are able to show
that all irreducible characters of \UT_n(q) with degree are
Kirillov functions. We also discuss some related results concerning the problem
of counting the irreducible constituents of individual supercharacters of
\UT_n(q).Comment: 34 pages, 5 table
Strong forms of linearization for Hopf monoids in species
A vector species is a functor from the category of finite sets with
bijections to vector spaces; informally, one can view this as a sequence of
-modules. A Hopf monoid (in the category of vector species) consists of a
vector species with unit, counit, product, and coproduct morphisms satisfying
several compatibility conditions, analogous to a graded Hopf algebra. We say
that a Hopf monoid is strongly linearized if it has a "basis" preserved by its
product and coproduct in a certain sense. We prove several equivalent
characterizations of this property, and show that any strongly linearized Hopf
monoid which is commutative and cocommutative possesses four bases which one
can view as analogues of the classical bases of the algebra of symmetric
functions. There are natural functors which turn Hopf monoids into graded Hopf
algebras, and applying these functors to strongly linearized Hopf monoids
produces several notable families of Hopf algebras. For example, in this way we
give a simple unified construction of the Hopf algebras of superclass functions
attached to the maximal unipotent subgroups of three families of classical
Chevalley groups.Comment: 35 pages; v2: corrected some typos, fixed attribution for Theorem
5.4.4; v3: some corrections, slight revisions, added references; v4: updated
references, numbering of results modified to conform with published version,
final versio
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