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In this paper we compute the explicit value of the standard symmetrizing form of the cyclotomic Hecke algebra of type $G(\ell,1,n)$ on each Murphy basis element. We do this for both the non-degenerate cyclotomic Hecke algebra of type…
We obtain a formula for the degrees of the varieties parameterizing complex algebraic curves of any divisor class and genus on P^2_6, the projective plane blown-up at 6 generic points. Moreover, the formula computes the degrees of the…
We derive a closed formula for the tensor product of a family of mixed tensors using Deligne's interpolating category $Rep(GL_{0})$. We use this formula to compute the tensor product between any two maximal atypical irreducible…
Basing on properties of the Mellin transform and Ramanujan's identities, which represent a ratio of products of Riemann's zeta- functions of different arguments in terms of the Dirichlet series of arithmetic functions, we obtain a number of…
For the standard symplectic forms on Jacobi and CMV matrices, we compute Poisson brackets of OPRL and OPUC, and relate these to other basic Poisson brackets and to Jacobians of basic changes of variable.
In the paper, the authors find two closed forms involving the Stirling numbers of the second kind and in terms of a determinant of combinatorial numbers for the Bernoulli polynomials and numbers.
In this note, we rederive quantum Pieri's formula and the rim hook algorithm in quantum Schubert calculus by studying multiplication in the equivariant cohomology ring of Grassmannians with respect to equivariant Schubert classes which are…
The author presents a computer implementation, calculating the terms of the Saneblidze-Umble diagonals on the permutahedron and the associahedron. The code is analyzed for correctness and presented in the paper, the source code of which…
In this text, We compute the equivariant cohomology of Bott-Samelson varieties. Thanks to this computation, we give a new demonstration for the formulas proved by Sarah Billey for the equivariant cohomology of Schubert varieties.
For any Shimura variety of Hodge type with hyperspecial level at a prime $p$ and automorphic lisse sheaf on it, we prove a formula, conjectured by Kottwitz \cite{Kottwitz90}, for the Lefschetz numbers of Frobenius-twisted Hecke…
We employ a dispersion relation that allows to recover the phase of the electromagnetic form factor of the pion from its absolute value above threshold. Compared to alternative approaches building on the phase, this approach builds on…
Let $(G_k)_{k\in\mathbb Z}$ be any sequence obeying the recurrence relation of the Fibonacci numbers. We derive formulas for $\sum_{j=1}^n{G_{j + t}^6}$ and $\sum_{j=1}^n{(-1)^{j - 1}G_{j + t}^5(G_{j + t - 1} + G_{j + t + 1})}$, thereby…
We establish supercongruences for two kinds of Ap\'ery-like numbers, which involve Bernoulli numbers and Bernoulli polynomials. Conjectural supercongruences of the same type for another four kinds of Ap\'ery-like numbers are also proposed.
We present a Poisson formula for sparse resultants and a formula for the product of the roots of a family of Laurent polynomials, which are valid for arbitrary families of supports. To obtain these formulae, we show that the sparse…
Adriano Garsia has provided an explicit formula to enumerate a certain class of permutations in the symmetric group S_n. In this short note, we prove a stronger version of the assertion under a specific conjecture.
In this paper we give a simple Torelli type theorem for curves of genus 6 and 8 by showing that these curves can be reconstructed from their Brill-Noether varieties. Among other results, it is shown that the focal variety of a general,…
We obtain a new coding and decoding method using the generalized Pell $(p,i)$ -numbers. The relations among the code matrix elements, error detection and correction have been established for this coding theory. We give two new blocking…
We present a new geometric proof of Pieri's formula, exhibiting an explicit chain of rational equivalences from a suitable sum of distinct Schubert varieties to the intersection of a Schubert variety with a special Schubert variety. The…
E-variables are tools for retaining type-I error guarantee with optional stopping. We extend E-variables for sequential two-sample tests to general null hypotheses and anytime-valid confidence sequences. We provide implementations for…
We give an explicit Pieri formula for Macdonald polynomials attached to the root system C_n (with equal multiplicities). By inversion we obtain an explicit expansion for two-row Macdonald polynomials of type C.