We classify the \fully tight" simply-laced Coxeter groups, that is, the ones whose iji-avoiding Kazhdan{Lusztig basis elements are monomials in the generators B s i. We then investigate the basis of the Temperley{Lieb algebra arising from the Kazhdan{Lusztig basis of the associated Hecke algebra, and prove that the basis coincides with the usual (monomial) basis.

We study a vertex operator algebra (VOA) V related to the M(3, p) Virasoro minimal series. This VOA reduces in the simplest case p = 4 to the level two integrable vacuum module of ŝl2. On V there is an action of a commutative current a(z), which is an analog of the current e(z) of ŝl2. Our main concern is the subspace W generated by this action from the highest weight vector of V . Using the Fo...

In this short follow-up of [3] we answer some questions posed there. In particular, we characterize two natural classes of linear operators acting on R[x1, x2, ..., xn] which (i) preserve the set of positive n-variate polynomials; (ii) either act diagonally in the monomial basis of R[x1, x2, ..., xn] or are translation-invariant.

We study a vertex operator algebra (VOA) V related to the M(3, p) Virasoro minimal series. This VOA reduces in the simplest case p = 4 to the level two integrable vacuum module of ŝl2. On V there is an action of a commutative current a(z), which is an analog of the current e(z) of ŝl2. Our main concern is the subspace W generated by this action from the highest weight vector of V . Using the Fo...

We obtain a geometric construction of a “standard monomial basis” for the homogeneous coordinate ring associated with any ample line bundle on any flag variety. This basis is compatible with Schubert varieties, opposite Schubert varieties, and unions of intersections of these varieties. Our approach relies on vanishing theorems and a degeneration of the diagonal; it also yields a standard monom...

We consider the ring of coinvariants for modular representations of cyclic groups of prime order. For all cases for which explicit generators for the ring of invariants are known, we give a reduced Gröbner basis for the Hilbert ideal and the corresponding monomial basis for the coinvariants. We also describe the decomposition of the coinvariants as a module over the group ring. For one family o...

We discuss efficient conversion algorithms for orthogonal polynomials. We describe a known conversion algorithm from an arbitrary orthogonal basis to the monomial basis, and deduce a new algorithm of the same complexity for the converse operation.

Using the Beilinson-Lusztig-MacPherson construction of the quantized enveloping algebra of gln and its associated monomial basis, we investigate q-Schur algebras Sq(n, r) as “little quantum groups”. We give a presentation for Sq(n, r) and obtain a new basis for the integral q-Schur algebra Sq(n, r), which consists of certain monomials in the original generators. Finally, when n > r, we interpre...

We show that the action of classical operators associated to the Mac-donald polynomials on the basis of Schur functions, S λ [X(t − 1)/(q − 1)], can be reduced to addition in λ−rings. This provides explicit formulas for the Macdonald polynomials expanded in this basis as well as in the ordinary Schur basis, S λ [X], and the monomial basis, m λ [X].