We determine the Betti numbers of the Springer fibers in type
A. To do this, we construct a cell decomposition of the Springer fibers. The codimension of the cells is given by an analogue of the ...Coxeter length. This makes our cell decomposition well suited for the calculation of Betti numbers.
In this note, we formulate and prove a branching rule for simple polynomial modules of the Lie superalgebra
. Our branching rules depend on the conjugacy class of a Borel subalgebra. A ...Gelfand-Tsetlin basis of a polynomial module associated to each Borel subalgebra is obtained in terms of generalized semistandard tableaux.
We investigate the large deviations of the shape of the random RSK Young diagrams associated with a random word of size n whose letters are independently drawn from an alphabet of size m = m(n). When ...the letters are drawn uniformly and when both n and m converge together to infinity, m not growing too fast with respect to n, the large deviations of the shape of the Young diagrams are shown to be the same as that of the spectrum of the traceless GUE. In the non-uniform case, a control of both highest probabilities will ensure that the length of the top row of the diagram satisfies a large deviation principle. In either case, both speeds and rate functions are identified. To complete our study, non-asymptotic concentration bounds for the length of the top row of the diagrams, that is, for the length of the longest increasing subsequence of the random word are also given for both models.
Super Jack-Laurent Polynomials Sergeev, A. N.
Algebras and representation theory,
10/2018, Letnik:
21, Številka:
5
Journal Article
Recenzirano
Odprti dostop
Let
D
n
,
m
be the algebra of quantum integrals of the deformed Calogero-Moser-Sutherland problem corresponding to the root system of the Lie superalgebra
𝔤
𝔩
(
n
,
m
)
. The algebra
D
n
,
m
acts ...naturally on the quasi-invariant Laurent polynomials and we investigate the corresponding spectral decomposition. Even for general value of the parameter
k
the spectral decomposition is not multiplicity free and we prove that the image of the algebra
D
n
,
m
in the algebra of endomorphisms of the generalised eigenspace is
k
ε
⊗
r
where
k
ε
is the algebra of dual numbers. The corresponding representation is the regular representation of the algebra
k
ε
⊗
r
.
In this article we have illustrated one solved problem in chemistry, which left room for improvements. This instance is raising an issue when a solved problem is to be considered solved. As we will ...see by using novel tools, not known at the time of solving a problem, one can arrived at additional unknown information on the problem. We will consider several problems of chemistry, some even considered as having no exact solution, which have been solved by using previously unknown concepts in chemistry. The last problem which we consider has been solved, but the problem is still open, as it might have additional solutions.
Jack polynomials generalize several classical families of symmetric polynomials, including Schur polynomials, and are further generalized by Macdonald polynomials. In 1989, Richard Stanley ...conjectured that if the Littlewood–Richardson coefficient for a triple of Schur polynomials is 1, then the corresponding coefficient for Jack polynomials can be expressed as a product of weighted hooks of the Young diagrams associated with the partitions indexing the coefficient. We prove a special case of this conjecture in which the partitions indexing the Littlewood–Richardson coefficient have at most three parts. We also show that this result extends to Macdonald polynomials.
We show that the generalized Bernstein bases in Müntz spaces defined by Hirschman and Widder (1949) and extended by Gelfond (1950) can be obtained as pointwise limits of the Chebyshev–Bernstein bases ...in Müntz spaces with respect to an interval a,1 as the positive real number a converges to zero. Such a realization allows for concepts of curve design such as de Casteljau algorithm, blossom, dimension elevation to be transferred from the general theory of Chebyshev blossoms in Müntz spaces to these generalized Bernstein bases that we termed here as Gelfond–Bernstein bases. The advantage of working with Gelfond–Bernstein bases lies in the simplicity of the obtained concepts and algorithms as compared to their Chebyshev–Bernstein bases counterparts.
► Show that Gelfond–Bernstein bases in Müntz spaces are limit of Chebyshev–Bernstein bases. ► Give a Schur function expression of Gelfond–Bernstein bases. ► Define the blossom of Gelfond–Bézier curves. ► Derive the de Casteljau algorithm in Müntz spaces. ► Show the conditions for the convergence of the dimension elevation algorithm to the Gelfond–Bézier curve.
We determine the most general form of a smooth function on Young diagrams, that is, a polynomial in the interlacing or multirectangular coordinates whose value depends only on the shape of the ...diagram. We prove that the algebra of such functions is isomorphic to quasi-symmetric functions, and give a noncommutative analog of this result.
We study asymptotics of an irreducible representation of the symmetric group
S
n
corresponding to a balanced Young diagram
λ (a Young diagram with at most
C
n
rows and columns for some fixed constant
...C) in the limit as
n tends to infinity. We show that there exists a constant
D (which depends only on
C) with a property that
|
χ
λ
(
π
)
|
=
|
Tr
ρ
λ
(
π
)
Tr
ρ
λ
(
e
)
|
⩽
(
D
max
(
1
,
|
π
|
2
n
)
n
)
|
π
|
,
where
|
π
|
denotes the length of a permutation (the minimal number of factors necessary to write
π as a product of transpositions). Our main tool is an analogue of the Frobenius character formula which holds true not only for cycles but for arbitrary permutations.
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