145 research outputs found
Hook formulas for skew shapes III. Multivariate and product formulas
We give new product formulas for the number of standard Young tableaux of
certain skew shapes and for the principal evaluation of the certain Schubert
polynomials. These are proved by utilizing symmetries for evaluations of
factorial Schur functions, extensively studied in the first two papers in the
series "Hook formulas for skew shapes" [arxiv:1512.08348, arxiv:1610.04744]. We
also apply our technology to obtain determinantal and product formulas for the
partition function of certain weighted lozenge tilings, and give various
probabilistic and asymptotic applications.Comment: 40 pages, 17 figures. This is the third paper in the series "Hook
formulas for skew shapes"; v2 added reference to [KO1] (arxiv:1409.1317)
where the formula in Corollary 1.1 had previously appeared; v3 Corollary 5.10
added, resembles published versio
Schur polynomials, banded Toeplitz matrices and Widom's formula
We prove that for arbitrary partitions and integers the sequence of Schur
polynomials for sufficiently large, satisfy a
linear recurrence. The roots of the characteristic equation are given
explicitly. These recurrences are also valid for certain sequences of minors of
banded Toeplitz matrices.
In addition, we show that Widom's determinant formula from 1958 is a special
case of a well-known identity for Schur polynomials
Skew Howe duality and limit shapes of Young diagrams
We consider the skew Howe duality for the action of certain dual pairs of Lie
groups on the exterior algebra as a probability measure on Young diagrams by the
decomposition into the sum of irreducible representations. We prove a
combinatorial version of this skew Howe for the pairs , ,
, and using crystal bases, which allows us to interpret the skew
Howe duality as a natural consequence of lattice paths on lozenge tilings of
certain partial hexagonal domains. The -representation multiplicity is
given as a determinant formula using the Lindstr\"om-Gessel-Viennot lemma and
as a product formula using Dodgson condensation. These admit natural
-analogs that we show equals the -dimension of a -representation (up
to an overall factor of ), giving a refined version of the combinatorial
skew Howe duality. Using these product formulas (at ), we take the
infinite rank limit and prove the diagrams converge uniformly to the limit
shape.Comment: 54 pages, 12 figures, 2 tables; v2 fixed typos in Theorem 4.10, 4.14,
shorter proof of Theorem 4.6 (thanks to C. Krattenthaler), proved of
Conjecture 4.17 in v
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