1,302 research outputs found
Anti-Power -fixes of the Thue-Morse Word
Recently, Fici, Restivo, Silva, and Zamboni introduced the notion of a
-anti-power, which is defined as a word of the form , where are distinct words of the
same length. For an infinite word and a positive integer , define
to be the set of all integers such that is a -anti-power, where denotes the -th letter of .
Define also ,
where denotes the Thue-Morse word. For all ,
is a well-defined positive integer,
and for sufficiently large, is a well-defined odd positive
integer. In his 2018 paper, Defant shows that and
grow linearly in . We generalize Defant's methods to prove that
and grow linearly in for any nonnegative
integer . In particular, we show that and . Additionally, we show
that and
.Comment: 19 page
Differential posets and restriction in critical groups
In recent work, Benkart, Klivans, and Reiner defined the critical group of a
faithful representation of a finite group , which is analogous to the
critical group of a graph. In this paper we study maps between critical groups
induced by injective group homomorphisms and in particular the map induced by
restriction of the representation to a subgroup. We show that in the abelian
group case the critical groups are isomorphic to the critical groups of a
certain Cayley graph and that the restriction map corresponds to a graph
covering map. We also show that when is an element in a differential tower
of groups, critical groups of certain representations are closely related to
words of up-down maps in the associated differential poset. We use this to
generalize an explicit formula for the critical group of the permutation
representation of the symmetric group given by the second author, and to
enumerate the factors in such critical groups.Comment: 18 pages; v2: minor edits and updated reference
Balance constants for Coxeter groups
The - Conjecture, originally formulated in 1968, is one of the
best-known open problems in the theory of posets, stating that the balance
constant (a quantity determined by the linear extensions) of any non-total
order is at least . By reinterpreting balance constants of posets in terms
of convex subsets of the symmetric group, we extend the study of balance
constants to convex subsets of any Coxeter group. Remarkably, we conjecture
that the lower bound of still applies in any finite Weyl group, with new
and interesting equality cases appearing.
We generalize several of the main results towards the - Conjecture
to this new setting: we prove our conjecture when is a weak order interval
below a fully commutative element in any acyclic Coxeter group (an
generalization of the case of width-two posets), we give a uniform lower bound
for balance constants in all finite Weyl groups using a new generalization of
order polytopes to this context, and we introduce generalized semiorders for
which we resolve the conjecture.
We hope this new perspective may shed light on the proper level of generality
in which to consider the - Conjecture, and therefore on which methods
are likely to be successful in resolving it.Comment: 27 page
On the Sperner property for the absolute order on complex reflection groups
Two partial orders on a reflection group, the codimension order and the
prefix order, are together called the absolute order when they agree. We show
that in this case the absolute order on a complex reflection group has the
strong Sperner property, except possibly for the Coxeter group of type ,
for which this property is conjectural. The Sperner property had previously
been established for the noncrossing partition lattice , a certain
maximal interval in the absolute order, but not for the entire poset, except in
the case of the symmetric group. We also show that neither the codimension
order nor the prefix order has the Sperner property for general complex
reflection groups.Comment: 12 pages, comments welcome; v2: minor edits and journal referenc
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