The algebra Sym of symmetric functions is a proper subalgebra of QSym: for example, M11 and M12 +M21 are symmetric, but M12 is not. As an algebra, QSym is generated by those monomial symmetric functions corresponding to Lyndon words in the positive integers [11, 6]. The subalgebra of QSym ⊂ QSym generated by all Lyndon words other than M1 has the vector space basis consisting of all monomial sy...

The symplectic group branching algebra, B, is a graded algebra whose components encode the multiplicities of irreducible representations of Sp2n−2(C) in each finite-dimensional irreducible representation of Sp2n(C). By describing on B an ASL structure, we construct an explicit standard monomial basis of B consisting of Sp2n−2(C) highest weight vectors. Moreover, B is known to carry a canonical ...

We develop a new approach to cluster algebras based on the notion of an upper cluster algebra, defined as an intersection of Laurent polynomial rings. Strengthening the Laurent phenomenon established in [6], we show that, under an assumption of “acyclicity”, a cluster algebra coincides with its “upper” counterpart, and is finitely generated; in this case, we also describe its defining ideal, an...

Let U be the quantized enveloping algebra associated to a simple Lie algebra g by Drinfel’d and Jimbo. Let λ be a classical fundamental weight for g, and V (λ) the irreducible, finite-dimensional type 1 highest weight U -module with highest weight λ. We show that the canonical basis for V (λ) (see Kashiwara [6, §0] and Lusztig [18, 14.4.12]) and the standard monomial basis (see [11, §§2.4 & 2.5...

We develop a new approach to cluster algebras based on the notion of an upper cluster algebra, defined as an intersection of Laurent polynomial rings. Strengthening the Laurent phenomenon established in [6], we show that, under an assumption of “acyclicity”, a cluster algebra coincides with its “upper” counterpart, and is finitely generated; in this case, we also describe its defining ideal, an...

The floating-point implementation of a function on an interval often reduces to polynomial approximation, the polynomial being typically provided by Remez algorithm. However, the floating-point evaluation of a Remez polynomial sometimes leads to catastrophic cancellations. This happens when some of the polynomial coefficients are very small in magnitude with respects to others. In this case, it...

We establish a generalization of the p-adic local monodromy theorem (of André, Mebkhout, and the author) in which differential equations on rigid analytic annuli are replaced by differential equations on so-called “fake annuli”. The latter correspond loosely to completions of a Laurent polynomial ring with respect to a valuation, which in this paper is restricted to be of monomial form; we defe...

We develop a new approach to cluster algebras based on the notion of an upper cluster algebra, defined as an intersection of Laurent polynomial rings. Strengthening the Laurent phenomenon established in [6], we show that, under an assumption of “acyclicity”, a cluster algebra coincides with its “upper” counterpart, and is finitely generated; in this case, we also describe its defining ideal, an...

We give a combinatorial proof of the factorization formula of modified Macdonald polynomials H̃λ(X ; q, t) when t is specialized at a primitive root of unity. Our proof is restricted to the special case where λ is a two columns partition. We mainly use the combinatorial interpretation of Haiman, Haglund and Loehr giving the expansion of H̃λ(X ; q, t) on the monomial basis.