Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The representation of the symmetric group on m-Tamari intervals   An m-ballot path of size n is a path on the square grid consisting of north
and east unit steps, starting at (0,0), ending at (mn,n), and never going below
the line {x=my}. The set of these paths can be equipped with a lattice
structure, called the m-Tamari lattice and denoted by T_n^{m}, which
generalizes the usual Tamari lattice T_n obtained when m=1. This lattice was
introduced by F. Bergeron in connection with the study of diagonal coinvariant
spaces in three sets of n variables. The representation of the symmetric group
S_n on these spaces is conjectured to be closely related to the natural
representation of S_n on (labelled) intervals of the m-Tamari lattice, which we
study in this paper. An interval [P,Q] of T_n^{m} is labelled if the north
steps of Q are labelled from 1 to n in such a way the labels increase along any
sequence of consecutive north steps. The symmetric group S_n acts on labelled
intervals of T_n^{m} by permutation of the labels. We prove an explicit
formula, conjectured by F. Bergeron and the third author, for the character of
the associated representation of S_n. In particular, the dimension of the
representation, that is, the number of labelled m-Tamari intervals of size n,
is found to be (m+1)^n(mn+1)^{n-2}. These results are new, even when m=1. The
form of these numbers suggests a connection with parking functions, but our
proof is not bijective. The starting point is a recursive description of
m-Tamari intervals. It yields an equation for an associated generating
function, which is a refined version of the Frobenius series of the
representation. This equation involves two additional variables x and y, a
derivative with respect to y and iterated divided differences with respect to
x. The hardest part of the proof consists in solving it, and we develop
original techniques to do so, partly inspired by previous work on polynomial
equations with "catalytic" variables.

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The categories of this paper are: physics
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-fin
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The categories of this paper are: nlin
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The categories of this paper are: plasm-ph
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The categories of this paper are: adap-org
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The categories of this paper are: atom-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: acc-phys
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The categories of this paper are: astro-ph
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The categories of this paper are: math-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-ph
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The categories of this paper are: econ
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The categories of this paper are: cond-mat