Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userShifted tableau switchings and shifted Littlewood-Richardson coefficients
286
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We provide two shifted analogues of the tableau switching process due to Benkart, Sottile, and Stroomer, the shifted tableau switching process and the modified shifted tableau switching process. They are performed by applying a sequence of specially contrived elementary transformations called {em switches} and turn out to have some spectacular properties. For instance, the maps induced from these algorithms are involutive and behave very nicely with respect to shifted Young tableaux whose reading words satisfy the lattice property. As an application, we give combinatorial interpretations of Schur $P$- and $Q$-function identities. We also demonstrate the relationship between the shifted tableau switching process and the shifted $J$-operation due to Worley.
rate research
Read More
We prove an identity for Littlewood--Richardson coefficients conjectured by Pelletier and Ressayre (arXiv:2005.09877). The proof relies on a novel birational involution defined over any semifield.
In this paper we explicitly compute all Littlewood-Richardson coefficients for semisimple or Kac-Moody groups G, that is, the structure coefficients of the cohomology algebra H^*(G/P), where P is a parabolic subgroup of G. These coefficients are of importance in enumerative geometry, algebraic combinatorics and representation theory. Our formula for the Littlewood-Richardson coefficients is given in terms of the Cartan matrix and the Weyl group of G. However, if some off-diagonal entries of the Cartan matrix are 0 or -1, the formula may contain negative summands. On the other hand, if the Cartan matrix satisfies $a_{ij}a_{ji}ge 4$ for all $i,j$, then each summand in our formula is nonnegative that implies nonnegativity of all Littlewood-Richardson coefficients. We extend this and other results to the structure coefficients of the T-equivariant cohomology of flag varieties G/P and Bott-Samelson varieties Gamma_ii(G).
Answering a question raised by S. Friedland, we show that the possible eigenvalues of Hermitian matrices (or compact operators) A, B, and C with C <= A + B are given by the same inequalities as in Klyachkos theorem for the case where C = A + B, except that the equality corresponding to tr(C) = tr(A) + tr(B) is replaced by the inequality corresponding to tr(C) <= tr(A) + tr(B). The possible types of finitely generated torsion modules A, B, and C over a discrete valuation ring such that there is an exact sequence B -> C -> A are characterized by the same inequalities.
Let $f(n)$ be a multiplicative function with $|f(n)|leq 1, q$ be a prime number and $a$ be an integer with $(a, q)=1, chi$ be a non-principal Dirichlet character modulo $q$. Let $varepsilon$ be a sufficiently small positive constant, $A$ be a large constant, $q^{frac12+varepsilon}ll Nll q^A$. In this paper, we shall prove that $$ sum_{nleq N}f(n)chi(n+a)ll Nfrac{loglog q}{log q} $$ and that $$ sum_{nleq N}f(n)chi(n+a_1)cdotschi(n+a_t)ll Nfrac{loglog q}{log q}, $$ where $tgeq 2, a_1, ldots, a_t$ are distinct integers modulo $q$.
Sulanke and Xin developed a continued fraction method that applies to evaluate Hankel determinants corresponding to quadratic generating functions. We use their method to give short proofs of Ciglers Hankel determinant conjectures, which were proved recently by Chang-Hu-Zhang using direct determinant computation. We find that shifted periodic continued fractions arise in our computation. We also discover and prove some new nice Hankel determinants relating to lattice paths with step set ${(1,1),(q,0), (ell-1,-1)}$ for integer parameters $m,q,ell$. Again shifted periodic continued fractions appear.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
