7,355 research outputs found
Simultaneous core partitions: parameterizations and sums
Fix coprime . We re-prove, without Ehrhart reciprocity, a conjecture
of Armstrong (recently verified by Johnson) that the finitely many simultaneous
-cores have average size , and that the
subset of self-conjugate cores has the same average (first shown by
Chen--Huang--Wang). We similarly prove a recent conjecture of Fayers that the
average weighted by an inverse stabilizer---giving the "expected size of the
-core of a random -core"---is . We also prove
Fayers' conjecture that the analogous self-conjugate average is the same if
is odd, but instead if is even. In principle,
our explicit methods---or implicit variants thereof---extend to averages of
arbitrary powers.
The main new observation is that the stabilizers appearing in Fayers'
conjectures have simple formulas in Johnson's -coordinates parameterization
of -cores.
We also observe that the -coordinates extend to parameterize general
-cores. As an example application with , we count the number of
-cores for coprime , verifying a recent conjecture of
Amdeberhan and Leven.Comment: v4: updated references to match final EJC versio
Decomposition matrices for low rank unitary groups
We study the decomposition matrices for the unipotent -blocks of finite
special unitary groups SU for unitary primes larger than . Up
to very few unknown entries, we give a complete solution for .
We also prove a general result for two-column partitions when divides
. This is achieved using projective modules coming from the -adic
cohomology of Deligne--Lusztig varieties
Parafermionic quasi-particle basis and fermionic-type characters
A new basis of states for highest-weight modules in \ZZ_k parafermionic
conformal theories is displayed. It is formulated in terms of an effective
exclusion principle constraining strings of fundamental parafermionic
modes. The states of a module are then built by a simple filling process, with
no singular-vector subtractions. That results in fermionic-sum representations
of the characters, which are exactly the Lepowsky-Primc expressions. We also
stress that the underlying combinatorics -- which is the one pertaining to the
Andrews-Gordon identities -- has a remarkably natural parafermionic
interpretation.Comment: minor modifications and proof in app. C completed; 34 pages (harvmac
b
Forbidden ordinal patterns in higher dimensional dynamics
Forbidden ordinal patterns are ordinal patterns (or `rank blocks') that
cannot appear in the orbits generated by a map taking values on a linearly
ordered space, in which case we say that the map has forbidden patterns. Once a
map has a forbidden pattern of a given length , it has forbidden
patterns of any length and their number grows superexponentially
with . Using recent results on topological permutation entropy, we study in
this paper the existence and some basic properties of forbidden ordinal
patterns for self maps on n-dimensional intervals. Our most applicable
conclusion is that expansive interval maps with finite topological entropy have
necessarily forbidden patterns, although we conjecture that this is also the
case under more general conditions. The theoretical results are nicely
illustrated for n=2 both using the naive counting estimator for forbidden
patterns and Chao's estimator for the number of classes in a population. The
robustness of forbidden ordinal patterns against observational white noise is
also illustrated.Comment: 19 pages, 6 figure
General runner removal and the Mullineux map
We prove a new `runner removal theorem' for -decomposition numbers of the
level 1 Fock space of type , generalising earlier theorems of
James--Mathas and the author. By combining this with another theorem relating
to the Mullineux map, we show that the problem of finding all -decomposition
numbers indexed by partitions of a given weight is a finite computation.Comment: 40 page
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