We study the family of ideals defined by mixed size minors of two-sided ladders of indeterminates. We compute their Gröbner bases with respect to a skewdiagonal monomial order, then we use them to compute the height of the ideals. We show that these ideals correspond to a family of irreducible projective varieties, that we call mixed ladder determinantal varieties. We show that these varieties ...

Due to the elimination property held by the lexicographic monomial order, the corresponding Gröbner bases display strong structural properties from which meaningful informations can easily be extracted. We study these properties for radical ideals of (co)dimension zero. The proof presented relies on a combinatorial decomposition of the finite set of points whereby iterated Lagrange interpolatio...

In this article, we consider the matrix equation $AXB=C$, where A, B, C are given matrices and give new necessary and sufficient conditions for the existence of the diagonal solutions and monomial solutions to this equation. We also present a general form of such solutions. Moreover, we consider the least squares problem $min_X |C-AXB |_F$ where $X$ is a diagonal or monomial matrix. The explici...

Let L be the generalized mixed product ideal induced by a monomial ideal I. In this paper we compute powers of the genearlized mixed product ideals and show that Lk  have a linear resolution if and only if Ik have a linear resolution for all k. We also introduce the generalized mixed polymatroidal ideals and prove that powers and monomial localizations of a generalized mixed polymatroidal ideal...

The theory of Nakajima monomials is a combinatorial scheme for realizing crystal bases of quantum groups. Nakajima introduced a certain set of monomials realizing the irreducible highest weight crystals in [16]. Kashiwara and Nakajima independently defined a crystal structure on the set of Nakajima monomials and also gave a realization of irreducible highest weight crystal B(λ) in terms of Naka...

We solve the long-standing problem of operator basis construction for fields with all masses and spins. Based on on-shell method, we propose a novel method to systematically construct complete set lowest dimensional amplitude bases at any given dimension through semistandard Young tableaus Lorentz subgroup $SU(2{)}_{r}$ global symmetry $U(N)$ ($N$ is number external legs), which can be directly...

We introduce an affine Schur algebra via the Hecke associated to Weyl group of type C. establish multiplication formulas on and algebra. Then we construct monomial bases canonical for The formula allows us a stabilization property family algebras that leads modified version ${\mathbf K}^{\mathfrak c}_n$. show c}_n$ is coideal subalgebra quantum ${\bf U}(\hat{\mathfrak{gl}}_n)$, $\big({\mathbf U...

It is well known that in the computation of Gröbner bases an arbitrarily small perturbation in the coefficients of polynomials may lead to a completely different staircase even if the roots of the polynomials change continuously. This phenomenon is called pseudo singularity in this paper. We show how such phenomenon may be detected and even “repaired” by adding a new variable and a binomial rel...

Very recently, we proposed the row-monomial distributed orthogonal space-time block codes (DOSTBCs) and showed that the row-monomial DOSTBCs achieved approximately twice higher bandwidth efficiency than the repetitionbased cooperative strategy [1]. However, we imposed two limitations on the row-monomial DOSTBCs. The first one was that the associated matrices of the codes must be row-monomial. T...