179 research outputs found
Schur Q-functions and degeneracy locus formulas for morphisms with symmetries
We give closed-form formulas for the fundamental classes of degeneracy loci
associated with vector bundle maps given locally by (not necessary square)
matrices which are symmetric (resp. skew-symmetric) w.r.t. the main diagonal.
Our description uses essentially Schur Q-polynomials of a bundle, and is based
on a certain push-forward formula for these polynomials in a Grassmann bundle.Comment: 22 pages, AMSTEX, misprints corrected, exposition improved. to appear
in the Proceedings of Intersection Theory Conference in Bologna, "Progress in
Mathematics", Birkhause
Interval structure of the Pieri formula for Grothendieck polynomials
We give a combinatorial interpretation of a Pieri formula for double
Grothendieck polynomials in terms of an interval of the Bruhat order. Another
description had been given by Lenart and Postnikov in terms of chain
enumerations. We use Lascoux's interpretation of a product of Grothendieck
polynomials as a product of two kinds of generators of the 0-Hecke algebra, or
sorting operators. In this way we obtain a direct proof of the result of Lenart
and Postnikov and then prove that the set of permutations occuring in the
result is actually an interval of the Bruhat order.Comment: 27 page
Schur Partial Derivative Operators
A lattice diagram is a finite list L=((p_1,q_1),...,(p_n,q_n) of lattice
cells. The corresponding lattice diagram determinant is \Delta_L(X;Y)=\det \|
x_i^{p_j}y_i^{q_j} \|. These lattice diagram determinants are crucial in the
study of the so-called ``n! conjecture'' of A. Garsia and M. Haiman. The space
M_L is the space spanned by all partial derivatives of \Delta_L(X;Y). The
``shift operators'', which are particular partial symmetric derivative
operators are very useful in the comprehension of the structure of the M_L
spaces. We describe here how a Schur function partial derivative operator acts
on lattice diagrams with distinct cells in the positive quadrant.Comment: 8 pages, LaTe
The Algebra of Binary Search Trees
We introduce a monoid structure on the set of binary search trees, by a
process very similar to the construction of the plactic monoid, the
Robinson-Schensted insertion being replaced by the binary search tree
insertion. This leads to a new construction of the algebra of Planar Binary
Trees of Loday-Ronco, defining it in the same way as Non-Commutative Symmetric
Functions and Free Symmetric Functions. We briefly explain how the main known
properties of the Loday-Ronco algebra can be described and proved with this
combinatorial point of view, and then discuss it from a representation
theoretical point of view, which in turns leads to new combinatorial properties
of binary trees.Comment: 49 page
Formal Proof of SCHUR Conjugate Function
The main goal of our work is to formally prove the correctness of the key
commands of the SCHUR software, an interactive program for calculating with
characters of Lie groups and symmetric functions. The core of the computations
relies on enumeration and manipulation of combinatorial structures. As a first
"proof of concept", we present a formal proof of the conjugate function,
written in C. This function computes the conjugate of an integer partition. To
formally prove this program, we use the Frama-C software. It allows us to
annotate C functions and to generate proof obligations, which are proved using
several automated theorem provers. In this paper, we also draw on methodology,
discussing on how to formally prove this kind of program.Comment: To appear in CALCULEMUS 201
Origin and demographic history of the endemic Taiwan spruce (Picea morrisonicola)
Taiwan spruce (Picea morrisonicola) is a vulnerable conifer species endemic to the island of Taiwan. A warming climate and competition from subtropical tree species has limited the range of Taiwan spruce to the higher altitudes of the island. Using seeds sampled from an area in the central mountain range of Taiwan, 15 nuclear loci were sequenced in order to measure genetic variation and to assess the long-term genetic stability of the species. Genetic diversity is low and comparable to other spruce species with limited ranges such as Picea breweriana, Picea chihuahuana, and Picea schrenkiana. Importantly, analysis using approximate Bayesian computation (ABC) provides evidence for a drastic decline in the effective population size approximately 0.3–0.5 million years ago (mya). We used simulations to show that this is unlikely to be a false-positive result due to the limited sample used here. To investigate the phylogenetic origin of Taiwan spruce, additional sequencing was performed in the Chinese spruce Picea wilsonii and combined with previously published data for three other mainland China species, Picea purpurea, Picea likiangensis, and P. schrenkiana. Analysis of population structure revealed that P. morrisonicola clusters most closely with P. wilsonii, and coalescent analyses using the program MIMAR dated the split to 4–8 mya, coincidental to the formation of Taiwan. Considering the population decrease that occurred after the split, however, led to a much more recent origin
Double Schubert polynomials for the classical groups
For each infinite series of the classical Lie groups of type B,C or D, we
introduce a family of polynomials parametrized by the elements of the
corresponding Weyl group of infinite rank. These polynomials represent the
Schubert classes in the equivariant cohomology of the appropriate flag variety.
They satisfy a stability property, and are a natural extension of the (single)
Schubert polynomials of Billey and Haiman, which represent non-equivariant
Schubert classes. They are also positive in a certain sense, and when indexed
by maximal Grassmannian elements, or by the longest element in a finite Weyl
group, these polynomials can be expressed in terms of the factorial analogues
of Schur's Q- or P-functions defined earlier by Ivanov.Comment: 41 pages, 2 tables; comments welcom
Zero-one Schubert polynomials
We prove that if σ∈Sm is a pattern of w∈Sn, then we can express the Schubert polynomial w as a monomial times σ (in reindexed variables) plus a polynomial with nonnegative coefficients. This implies that the set of permutations whose Schubert polynomials have all their coefficients equal to either 0 or 1 is closed under pattern containment. Using Magyar's orthodontia, we characterize this class by a list of twelve avoided patterns. We also give other equivalent conditions on w being zero-one. In this case, the Schubert polynomial w is equal to the integer point transform of a generalized permutahedron
Subresultants in multiple roots: an extremal case
We provide explicit formulae for the coefficients of the order-d polynomial
subresultant of (x-\alpha)^m and (x-\beta)^n with respect to the set of
Bernstein polynomials \{(x-\alpha)^j(x-\beta)^{d-j}, \, 0\le j\le d\}. They are
given by hypergeometric expressions arising from determinants of binomial
Hankel matrices.Comment: 18 pages, uses elsart. Revised version accepted for publication at
Linear Algebra and its Application
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