3,793 research outputs found
Pieri's Formula for Generalized Schur Polynomials
Young's lattice, the lattice of all Young diagrams, has the
Robinson-Schensted-Knuth correspondence, the correspondence between certain
matrices and pairs of semi-standard Young tableaux with the same shape. Fomin
introduced generalized Schur operators to generalize the
Robinson-Schensted-Knuth correspondence. In this sense, generalized Schur
operators are generalizations of semi-standard Young tableaux. We define a
generalization of Schur polynomials as expansion coefficients of generalized
Schur operators. We show that the commutating relation of generalized Schur
operators implies Pieri's formula to generalized Schur polynomials
Applications of BGP-reflection functors: isomorphisms of cluster algebras
Given a symmetrizable generalized Cartan matrix , for any index , one
can define an automorphism associated with of the field of rational functions of independent indeterminates It is an isomorphism between two cluster algebras associated to the
matrix (see section 4 for precise meaning). When is of finite type,
these isomorphisms behave nicely, they are compatible with the BGP-reflection
functors of cluster categories defined in [Z1, Z2] if we identify the
indecomposable objects in the categories with cluster variables of the
corresponding cluster algebras, and they are also compatible with the
"truncated simple reflections" defined in [FZ2, FZ3]. Using the construction of
preprojective or preinjective modules of hereditary algebras by Dlab-Ringel
[DR] and the Coxeter automorphisms (i.e., a product of these isomorphisms), we
construct infinitely many cluster variables for cluster algebras of infinite
type and all cluster variables for finite types.Comment: revised versio
Cluster algebras and Grassmannians of type G2
We prove a conjecture of Geiss, Leclerc and Schr\"{o}er, producing cluster
algebra structures on multi-homogeneous coordinate ring of partial flag
varieties, for the case . As a consequence we sharpen the known fact that
coordinate ring of the double Bruhat cell is an upper cluster
algebra, by proving that it is a cluster algebra.Comment: minor typos correcte
On semiring complexity of Schur polynomials
Semiring complexity is the version of arithmetic circuit complexity that allows only two operations: addition and multiplication. We show that semiring complexity of a Schur polynomial {s_\lambda(x_1,\dots,x_k)} labeled by a partition {\lambda=(\lambda_1\ge\lambda_2\ge\cdots)} is bounded by {O(\log(\lambda_1))} provided the number of variables is fixed
Phonons in magnon superfluid and symmetry breaking field
Recent experiments [1],[2] which measured the spectrum of the Goldstone
collective mode of coherently precessing state in 3He-B are discussed using the
presentation of the coherent spin precession in terms of the Bose-Einstein
condensation of magnons. The mass in the spectrum of the Goldstone boson --
phonon in the superfluid magnon liquid -- is induced by the symmetry breaking
field, which is played by the RF magnetic fieldComment: 2 pages, JETP Letters style, no figures, version accepted in JETP
Letter
Q-systems, Heaps, Paths and Cluster Positivity
We consider the cluster algebra associated to the -system for as a
tool for relating -system solutions to all possible sets of initial data. We
show that the conserved quantities of the -system are partition functions
for hard particles on particular target graphs with weights, which are
determined by the choice of initial data. This allows us to interpret the
simplest solutions of the Q-system as generating functions for Viennot's heaps
on these target graphs, and equivalently as generating functions of weighted
paths on suitable dual target graphs. The generating functions take the form of
finite continued fractions. In this setting, the cluster mutations correspond
to local rearrangements of the fractions which leave their final value
unchanged. Finally, the general solutions of the -system are interpreted as
partition functions for strongly non-intersecting families of lattice paths on
target lattices. This expresses all cluster variables as manifestly positive
Laurent polynomials of any initial data, thus proving the cluster positivity
conjecture for the -system. We also give an alternative formulation in
terms of domino tilings of deformed Aztec diamonds with defects.Comment: 106 pages, 38 figure
Radiation of a relativistic electron with non-equilibrium own Coulomb field
The condition and specific features of the non-dipole regime of radiation is discussed in the context of the results of the recent CERN experiment NA63 on measurement of the radiation power spectrum of 149 GeV electrons in thin
tantalum targets. The first observation of a logarithmic dependence of radiation yield on the target thickness that was done there is the conclusive evidence of the effect of radiation suppression in a thin layer of matter, which was predicted many years ago, and which is the direct manifestation of the radiation of a relativistic electron with non-equilibrium own Coulomb field. The special features of the angular distribution of the radiation and its polarization in a thin target at non-dipole regime are proposed for a new experimental study
On the Optimality of Pseudo-polynomial Algorithms for Integer Programming
In the classic Integer Programming (IP) problem, the objective is to decide
whether, for a given matrix and an -vector , there is a non-negative integer -vector such that . Solving
(IP) is an important step in numerous algorithms and it is important to obtain
an understanding of the precise complexity of this problem as a function of
natural parameters of the input.
The classic pseudo-polynomial time algorithm of Papadimitriou [J. ACM 1981]
for instances of (IP) with a constant number of constraints was only recently
improved upon by Eisenbrand and Weismantel [SODA 2018] and Jansen and Rohwedder
[ArXiv 2018]. We continue this line of work and show that under the Exponential
Time Hypothesis (ETH), the algorithm of Jansen and Rohwedder is nearly optimal.
We also show that when the matrix is assumed to be non-negative, a
component of Papadimitriou's original algorithm is already nearly optimal under
ETH.
This motivates us to pick up the line of research initiated by Cunningham and
Geelen [IPCO 2007] who studied the complexity of solving (IP) with non-negative
matrices in which the number of constraints may be unbounded, but the
branch-width of the column-matroid corresponding to the constraint matrix is a
constant. We prove a lower bound on the complexity of solving (IP) for such
instances and obtain optimal results with respect to a closely related
parameter, path-width. Specifically, we prove matching upper and lower bounds
for (IP) when the path-width of the corresponding column-matroid is a constant.Comment: 29 pages, To appear in ESA 201
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