166 research outputs found
The generalized Delta conjecture at t=0
We prove the cases q=0 and t=0 of the generalized Delta conjecture of
Haglund, Remmel and Wilson involving the symmetric function
. Our theorem generalizes recent results by
Garsia, Haglund, Remmel and Yoo. This proves also the case q=0 of our recent
generalized Delta square conjecture.Comment: 21 pages, 3 figure
The Delta square conjecture
We conjecture a formula for the symmetric function
in terms of
decorated partially labelled square paths. This can be seen as a generalization
of the square conjecture of Loehr and Warrington (Loehr, Warrington 2007),
recently proved by Sergel (Sergel 2017) after the breakthrough of Carlsson and
Mellit (Carlsson, Mellit 2018). Moreover, it extends to the square case the
combinatorics of the generalized Delta conjecture of Haglund, Remmel and Wilson
(Haglund, Remmel, Wilson 2015), answering one of their questions. We support
our conjecture by proving the specialization , reducing it to the same
case of the Delta conjecture, and the Schr\"{o}der case, i.e. the case . The latter provides a broad generalization of the
-square theorem of Can and Loehr (Can, Loehr 2006). We give also a
combinatorial involution, which allows to establish a linear relation among our
conjectures (as well as the generalized Delta conjectures) with fixed and
. Finally, in the appendix, we give a new proof of the Delta conjecture at
.Comment: 27 pages, 6 figures. arXiv admin note: text overlap with
arXiv:1807.0541
Partial and global representations of finite groups
Given a subgroup H of a finite group G, we begin a systematic study of the
partial representations of G that restrict to global representations of H.
After adapting several results from [DEP00] (which correspond to the case where
H is trivial), we develop further an effective theory that allows explicit
computations. As a case study, we apply our theory to the symmetric group and
its subgroup of permutations fixing 1: this provides a natural extension of the
classical representation theory of the symmetric group.Comment: 31 page
-positivity of vertical strip LLT polynomials
In this article we prove the -positivity of when
is a vertical strip LLT polynomial. This property has
been conjectured by Alexandersson and Panova, and by Garsia, Haglund, Qiu and
Romero, and it implies several -positivities conjectured by them and also by
Bergeron.
We make use of a result of Carlsson and Mellit that shows that a vertical
strip LLT polynomial can be obtained by applying certain compositions of
operators of the Dyck path algebra to the constant . Our proof gives in fact
an algorithm to expand these symmetric functions in the elementary basis, and
it shows, as a byproduct, that these compositions of operators are actually
multiplication operators.Comment: 11 pages, 2 figure
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