A Littlewood-Richardson rule for factorial Schur functions

We give a combinatorial rule for calculating the coefficients in the expansion of a product of two factorial Schur functions. It is a special case of a more general rule which also gives the coefficients in the expansion of a skew factorial Schur function. Applications to Capelli operators and quantum immanants are also given.

     

A self paired Hopf algebra on double posets and a Littlewood-Richardson rule

Let D be the set of isomorphism types of finite double partially ordered sets, that is sets endowed with two partial orders. On ZD we define a product and a coproduct, together with an internal product, that is, degree-preserving. With these operations ZD is a Hopf algebra. We define a symmetric bilinear form on this Hopf algebra: it counts the number of pictures (in the sense of Zelevinsky) between two double posets. This form is a Hopf pairing, which means that product and coproduct are adjoint each to another. The product and coproduct correspond respectively to disjoint union of posets and to a natural decomposition of a poset into order ideals. Restricting to special double posets (meaning that the second order is total), we obtain a notion equivalent to Stanley's labelled posets, and a Hopf subalgebra already considered by Blessenohl and Schocker. The mapping which maps each double poset onto the sum of the linear extensions of its first order, identified via its second (total) order with permutations, is a Hopf algebra homomorphism, which is isometric and preserves the internal product, onto the Hopf algebra of permutations, previously considered by the two authors. Finally, the scalar product between any special double poset and double posets naturally associated to integer partitions is described by an extension of the Littlewood-Richardson rule.     

Generalised stretched Littlewood-Richardson coefficients

The Littlewood-Richardson (LR) coefficient counts, among many other things, the LR tableaux of a given shape and a given content. We prove that the number of LR tableaux weakly increases if one adds to its shape and content the shape and the content of another LR tableau. We also investigate the behaviour of the number of LR tableaux, if one repeatedly adds to the shape another shape with either fixed or arbitrary content. This is a generalisation of the stretched LR coefficients, where one repeatedly adds the same shape and content to itself.     

On the complexity of computing Kostka numbers and Littlewood-Richardson coefficients

Kostka numbers and Littlewood-Richardson coefficients appear in combinatorics and representation theory. Interest in their computation stems from the fact that they are present in quantum mechanical computations since Wigner [15]. In recent times, there have been a number of algorithms proposed to perform this task [1–3, 11, 12]. The issue of their computational complexity has received at-tention in the past, and was raised recently by E. Rassart in [11]. We prove that the problem of computing either quantity is #P-complete. Thus, unless P = NP, which is widely disbelieved, there do not exist efficient algorithms that compute these numbers.     

On applying the Littlewood-Richardson rule

We will consider the realization of the Littlewood-Richardson rule for the outer product of symmetric group characters using polynomial generating functions. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 1, pp. 199–213, 2007.     

A combinatorial proof of the reduction formula for Littlewood-Richardson coefficients

There are well-known reduction formulas for the universal Schubert coefficients defined on Grassmannians. These coefficients are also known as the Littlewood-Richardson coefficients in the theory of symmetric functions. We restate the reduction formulas combinatorially and provide a combinatorial proof for them.     

Factorisation of Littlewood-Richardson coefficients

The hive model is used to show that the saturation of any essential Horn inequality leads to the factorisation of Littlewood-Richardson coefficients. The proof is based on the use of combinatorial objects known as puzzles. These are shown not only to account for the origin of Horn inequalities, but also to determine the constraints on hives that lead to factorisation. Defining a primitive Littlewood-Richardson coefficient to be one for which all essential Horn inequalities are strict, it is shown that every Littlewood-Richardson coefficient can be expressed as a product of primitive coefficients. Precisely the same result is shown to apply to the polynomials defined by stretched Littlewood-Richardson coefficients.     

Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients

Let be a quiver without oriented cycles. For a dimension vector let be the set of representations of with dimension vector . The group acts on . In this paper we show that the ring of semi-invariants is spanned by special semi-invariants associated to representations of . From this we show that the set of weights appearing in is saturated. In the case of triple flag quiver this reduces to the results of Knutson and Tao on the saturation of the set of triples of partitions for which the Littlewood-Richardson coefficient is nonzero.

     

A bijection between Littlewood-Richardson tableaux and rigged configurations

We define a bijection from Littlewood-Richardson tableaux to rigged configurations and show that it preserves the appropriate statistics. This proves in particular a quasi-particle expression for the generalized Kostka polynomials labeled by a partition and a sequence of rectangles R. The generalized Kostka polynomials are q-analogues of multiplicities of the irreducible -module of highest weight in the tensor product .     

On almost one-to-one maps

A continuous map of topological spaces is said to be almost -to- if the set of the points such that is dense in ; it is said to be light if pointwise preimages are zero dimensional. We study almost 1-to-1 light maps of some compact and -compact spaces (e.g., -manifolds or dendrites) and prove that in some important cases they must be homeomorphisms or embeddings. In a forthcoming paper we use these results and show that if is a minimal self-mapping of a 2-manifold , then point preimages under are tree-like continua and either is a union of 2-tori, or is a union of Klein bottles permuted by .

     

On the Omori-Yau almost maximum principle

We introduce a new sufficient condition for the conclusion of the Omori-Yau almost maximum principle in terms of the existence of a special exhaustion function. This seems presenting perhaps the easiest proof. We also demonstrate that the existing sufficient conditions imply our sufficient condition.     

On the almost sure central limit theorem for randomly indexed sums

Let {S_n, n ≥ 1} be partial sums of independent identically distributed random variables. The almost sure version of CLT is generalized on the case of randomly indexed sums {S_Nn, n ≥ 1}, where {N_n, n ≥ 1} is a sequence of positive integer‐valued random variables independent of {S_n, n ≥ 1}. The affects of nonrandom centering and norming are considered too (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a Conjectured Formula for Quiver Varieties

In A.S. Buch and W. Fulton [Invent. Math. 135 (1999), 665–687] a formula for the cohomology class of a quiver variety is proved. This formula writes the cohomology class of a quiver variety as a linear combination of products of Schur polynomials. In the same paper it is conjectured that all of the coefficients in this linear combination are non-negative, and given by a generalized Littlewood-Richardson rule, which states that the coefficients count certain sequences of tableaux called factor sequences. In this paper I prove some special cases of this conjecture. I also prove that the general conjecture follows from a stronger but simpler statement, for which substantial computer evidence has been obtained. Finally I will prove a useful criterion for recognizing factor sequences.     

Left almost split morphisms and generalized weyl algebras

It is proved that the category of modules of finite length over a broad class of generalized Weyl algebras contains no left almost split morphism starting from a simple module. It is shown that a similar assertion holds for the algebraUsl₂(k) over an algebraically closed field k of characteristic 0. As a by-product, a new series of simple modules for such algebras is constructed. Translated fromMatematicheskie Zametki, Vol. 66, No. 5 pp. 734–740, November, 1999.     

On Weyl almost periodic selections of multivalued maps

We prove that Weyl almost periodic multivalued maps R∋t→F(t)∈clU have Weyl almost periodic selections, where clU is the collection of non-empty closed sets of a complete metric space U.     

On the positive almost periodic type solutions for some nonlinear delay integral equations

In this paper, we discuss the existence of positive almost periodic type solutions for some nonlinear delay integral equations, by constructing a new fixed point theorem in the cone. Some known results are extended.     

On the almost Gorenstein property of determinantal rings

In this paper, we investigate the question of when the determinantal ring R over a field k is an almost Gorenstein local/graded ring in the sense of [14 Goto, S., Takahashi, R., Taniguchi, N. (2015). Almost Gorenstein rings - towards a theory of higher dimension. J. Pure Appl. Algebra 219:2666–2712.[Crossref], [Web of Science ®] , [Google Scholar]]. As a consequence of the main result, we see that if R is a non-Gorenstein almost Gorenstein local/graded ring, then the ring R has a minimal multiplicity.