Littlewood-Richardson系数的组合学
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摘要
Littlewood-Richardson系数是组合数学中的一个重要研究对象,同时也是代数以及代数几何中的重要研究对象。在组合数学中,Littlewood-Richardson系数是斜Schur函数关于Schur函数的展开式中的系数。该类系数有多种组合解释,其中最著名的就是由Littlewood和Richardson于1934年提出的Littlewood-Richardson法则,即系数c_(μv)~λ等于形状为λ/μ,类型为v,并且反阅读字为格排列的半标准杨表的个数。在群表示理论中,Littlewood-Richardson系数给出了一般线性群的不可约表示φ~λ在φ~μ与φ~v的张量积中的重数。在代数几何理论中,它们是两个Schubert类乘积展开式中的系数。
本篇论文的主要结果是关于Littlewood-Richardson系数的一些组合性质及其在组合数学中的应用。首先,我们利用hive模型研究了Littlewood-Richardson系数的一些组合性质,并给出了所有单重的斜Schur函数的刻画,即在这类斜Schur函数关于Schur函数的展开式中,所有的系数均为0或者1.然后,我们通过研究组合数学中的一个q对数凸问题给出了Littlewood-Richardson系数的一个应用。我们证明了多项式序列(?)的q对数凸性质,其中Littlewood-Richardson系数的一个重要性质-对偶Pieri法则在证明过程中起到了重要的作用。
在第一章中,我们首先给出了对称函数,特别是Schur函数和Littlewood-Richardson系数的发展历史和背景知识,同时也介绍了q对数凸和q对数凹的背景知识,然后给出了本篇论文将要用到的相关记号和定义。
在第二章中,我们利用hive模型讨论了斜Schur函数展成Schur函数的表达式中Littlewood-Richardson系数为单重的问题。Hive模型是一个可以用来研究Littlewood-Richardson系数及其性质的组合工具,它是Littlewood-Richardson法则的又一种表现形式。借助hive模型,我们给出了斜Schur函数关于Schur函数的展开式为单重的充分必要条件。从证明过程中我们可以看到,与传统的Littlewood-Richardson法则相比,利用hive模型来讨论Littlewood-Richardson系数可以使得问题的组合根源更加清晰,体现了hive模型在研究Littlewood-Richardson系数方面的优势,特别是在证明本章主要定理中的必要条件时,hive模型使得证明更加直观。同时,通过hive模型我们还能更深入地研究展开式中Littlewood-Richardson系数不满足单重条件的一些内在原因。
在第三章中,我们利用Littlewood-Richardson系数的一些性质证明了王毅等人提出的关于多项式序列(?)的q对数凸性质的一个猜想,该方法给出了处理组合数学中q对数凸问题的一种新方法。该类多项式是集合[n]上类型B的不交划分的生成函数,同时也出现在根系格增长级数的相关理论中。对于集合[n]上类型A的不交划分的生成函数,即Narayana多项式,陈永川等人已经证明了它的q对数凸性质。通过利用Schur函数理论中的Pieri法则和Jacobi-Trudi恒等式,我们将一个关于初等对称函数乘积的和式展开成Schur函数,并证明了该展开式的Schur非负性,然后通过利用对称函数的主特殊化证明其q对数凸性质。同时,我们还证明了王毅等提出的该类多项式在任何线性变化下保持对数凸性质的一个充分条件,从而得到(?)的线性变换都是保持对数凸性质的。
最后,我们在附录中给出了一组关于Littlewood-Richardson系数的不等式的基于hive模型的组合证明。
Littlewood-Richardson coefficient is an important object in combinatorics, it also plays a crucial role in algebra and algebraic geometry. In combinatorics, Littlewood-Richardson coefficients arise as the coefficients in the expansion of a skew Schur function in terms of ordinary Schur functions. These coefficients have various combinatorial interpretations, the most well-known one is that the coefficient c_(μv)~λequals the number of semistandard Young tableaux of shapeλ/μand type v whose reverse reading word is a lattice permutation, which was first discovered by Littlewood and Richardson. In representation theory, they give the multiplicity of the irreducible polynomial representationsφ~λin the tensor product ofφ~μandφ~v. In the theory of algebraic geometry, they are just the coefficients in the product of two Schubert classes.
The main results of this thesis are about some properties and the applications of Littlewood-Richardson coefficients in combinatorics. First, we present some properties of Littlewood-Richardson coefficient by employing the method of hive model and give the characterization of the skew Schur functions which are multiplicity-free in their expansions in terms of Schur functions, that is, each of the coefficients in the resulting Schur function expansions is 0 or 1. Second, we demonstrate the applications of Littlewood-Richardson coefficients by studying a q-log-convexity problem. We prove a conjecture on the q-log-convexity of the polynomial sequence (?) in which an important property of Littlewood-Richardson coefficient known as the dual Pieri's rule plays an important role.
In Chapter 1, we first give a background on symmetric functions, especially on Schur functions and the Littlewood-Richardson rule, as well as a background on q-log-convexity and q-log-concavity. Then we introduce some notations and definitions which will be used in this thesis.
In Chapter 2, we discuss the multiplicity-free problem of skew Schur function expansions by virtue of the hive model. The hive model is a combinatorial device that may be used to determine Littlewood-Richardson coefficients and study their properties. It represents an alternative to the use of the Littlewood-Richardson rule. Resorting to this combinatorial tool, we give the sufficient and necessary conditions for a skew Schur function s_(λ/μ) being multiplicity-free. One may see a number of advantages of this hive model method, including the fact that it allows a direct proof that all the cases enumerated in our main result are indeed multiplicity-free. Also, the hive model offers some insight into the origin of the breakdown of multiplicity-freeness for expansions of skew Schur functions.
In Chapter 3, we employ some properties of Littlewood-Richardson coefficients to prove a conjecture of Liu and Wang on the q-log-convexity of the polynomial sequence (?). The polynomial is the generating function of the lattice of noncrossing partitions of type B on [n], and it also arises in the theory of growth series of the root lattice. For the generating function of the lattice of noncrossing partitions of type A on [n], which is known as the Narayana polynomial, Chen, Wang and Yang have proved its q-log-convexity. By using Pieri's rule and the Jacobi-Trudi identity for Schur functions, we obtain an expansion of a sum of products of elementary symmetric functions in terms of Schur functions with nonnegative coefficients. Then the principal specialization leads to the q-log-convexity. We also prove that a technical condition of Liu and Wang holds for the squares of the binomial coefficients. Hence we deduce that the linear transformation with respect to the triangular array (?) is log-convexity preserving.
Finally, we provide a hive model proof of a pair of inequalities on Littlewood-Richardson coefficients in the Appendix.
本篇论文的主要结果是关于Littlewood-Richardson系数的一些组合性质及其在组合数学中的应用。首先,我们利用hive模型研究了Littlewood-Richardson系数的一些组合性质,并给出了所有单重的斜Schur函数的刻画,即在这类斜Schur函数关于Schur函数的展开式中,所有的系数均为0或者1.然后,我们通过研究组合数学中的一个q对数凸问题给出了Littlewood-Richardson系数的一个应用。我们证明了多项式序列(?)的q对数凸性质,其中Littlewood-Richardson系数的一个重要性质-对偶Pieri法则在证明过程中起到了重要的作用。
在第一章中,我们首先给出了对称函数,特别是Schur函数和Littlewood-Richardson系数的发展历史和背景知识,同时也介绍了q对数凸和q对数凹的背景知识,然后给出了本篇论文将要用到的相关记号和定义。
在第二章中,我们利用hive模型讨论了斜Schur函数展成Schur函数的表达式中Littlewood-Richardson系数为单重的问题。Hive模型是一个可以用来研究Littlewood-Richardson系数及其性质的组合工具,它是Littlewood-Richardson法则的又一种表现形式。借助hive模型,我们给出了斜Schur函数关于Schur函数的展开式为单重的充分必要条件。从证明过程中我们可以看到,与传统的Littlewood-Richardson法则相比,利用hive模型来讨论Littlewood-Richardson系数可以使得问题的组合根源更加清晰,体现了hive模型在研究Littlewood-Richardson系数方面的优势,特别是在证明本章主要定理中的必要条件时,hive模型使得证明更加直观。同时,通过hive模型我们还能更深入地研究展开式中Littlewood-Richardson系数不满足单重条件的一些内在原因。
在第三章中,我们利用Littlewood-Richardson系数的一些性质证明了王毅等人提出的关于多项式序列(?)的q对数凸性质的一个猜想,该方法给出了处理组合数学中q对数凸问题的一种新方法。该类多项式是集合[n]上类型B的不交划分的生成函数,同时也出现在根系格增长级数的相关理论中。对于集合[n]上类型A的不交划分的生成函数,即Narayana多项式,陈永川等人已经证明了它的q对数凸性质。通过利用Schur函数理论中的Pieri法则和Jacobi-Trudi恒等式,我们将一个关于初等对称函数乘积的和式展开成Schur函数,并证明了该展开式的Schur非负性,然后通过利用对称函数的主特殊化证明其q对数凸性质。同时,我们还证明了王毅等提出的该类多项式在任何线性变化下保持对数凸性质的一个充分条件,从而得到(?)的线性变换都是保持对数凸性质的。
最后,我们在附录中给出了一组关于Littlewood-Richardson系数的不等式的基于hive模型的组合证明。
Littlewood-Richardson coefficient is an important object in combinatorics, it also plays a crucial role in algebra and algebraic geometry. In combinatorics, Littlewood-Richardson coefficients arise as the coefficients in the expansion of a skew Schur function in terms of ordinary Schur functions. These coefficients have various combinatorial interpretations, the most well-known one is that the coefficient c_(μv)~λequals the number of semistandard Young tableaux of shapeλ/μand type v whose reverse reading word is a lattice permutation, which was first discovered by Littlewood and Richardson. In representation theory, they give the multiplicity of the irreducible polynomial representationsφ~λin the tensor product ofφ~μandφ~v. In the theory of algebraic geometry, they are just the coefficients in the product of two Schubert classes.
The main results of this thesis are about some properties and the applications of Littlewood-Richardson coefficients in combinatorics. First, we present some properties of Littlewood-Richardson coefficient by employing the method of hive model and give the characterization of the skew Schur functions which are multiplicity-free in their expansions in terms of Schur functions, that is, each of the coefficients in the resulting Schur function expansions is 0 or 1. Second, we demonstrate the applications of Littlewood-Richardson coefficients by studying a q-log-convexity problem. We prove a conjecture on the q-log-convexity of the polynomial sequence (?) in which an important property of Littlewood-Richardson coefficient known as the dual Pieri's rule plays an important role.
In Chapter 1, we first give a background on symmetric functions, especially on Schur functions and the Littlewood-Richardson rule, as well as a background on q-log-convexity and q-log-concavity. Then we introduce some notations and definitions which will be used in this thesis.
In Chapter 2, we discuss the multiplicity-free problem of skew Schur function expansions by virtue of the hive model. The hive model is a combinatorial device that may be used to determine Littlewood-Richardson coefficients and study their properties. It represents an alternative to the use of the Littlewood-Richardson rule. Resorting to this combinatorial tool, we give the sufficient and necessary conditions for a skew Schur function s_(λ/μ) being multiplicity-free. One may see a number of advantages of this hive model method, including the fact that it allows a direct proof that all the cases enumerated in our main result are indeed multiplicity-free. Also, the hive model offers some insight into the origin of the breakdown of multiplicity-freeness for expansions of skew Schur functions.
In Chapter 3, we employ some properties of Littlewood-Richardson coefficients to prove a conjecture of Liu and Wang on the q-log-convexity of the polynomial sequence (?). The polynomial is the generating function of the lattice of noncrossing partitions of type B on [n], and it also arises in the theory of growth series of the root lattice. For the generating function of the lattice of noncrossing partitions of type A on [n], which is known as the Narayana polynomial, Chen, Wang and Yang have proved its q-log-convexity. By using Pieri's rule and the Jacobi-Trudi identity for Schur functions, we obtain an expansion of a sum of products of elementary symmetric functions in terms of Schur functions with nonnegative coefficients. Then the principal specialization leads to the q-log-convexity. We also prove that a technical condition of Liu and Wang holds for the squares of the binomial coefficients. Hence we deduce that the linear transformation with respect to the triangular array (?) is log-convexity preserving.
Finally, we provide a hive model proof of a pair of inequalities on Littlewood-Richardson coefficients in the Appendix.
引文
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[2] A.C. Aitken, Note on dual symmetric functions, Proc. Edinburgh Math. Soc.2 (1) (1931), 164-167.
[3] A.C. Aitken, The monomial expansion of determinantal symmetric functions,Proc. Royal Soc. Edinburgh 61 (A) (1943), 300-310.
[4] F. Ardila, M. Beck, S. Hosten, J. Pfeifie, and K. Seashore, Root polytopes and growth series of root lattices, arXiv:0809.5123vl.
[5] M. Benson, Growth series of finite extensions of (?)~n are rational, Invent.Math. 73 (1983), 251-269.
[6] F. Brenti, Log-concave and unimodal sequences in algebra, combinatorics,and geometry: an update, Contemp. Math. 178 (1994), 71-89.
[7] A.S. Buch, The saturation conjecture (after A.Knutson and T. Tao). With an appendix by W. Fulton, Enseign. Math. 46 (2000), 43-60.
[8] L.M. Butler, The q-log-concavity of q-binomial coefficients, J. Combin. Theory Ser. A 54 (1990), 54-63.
[9] A.D. Berenstein and A.V. Zelevinsky, Triple multiplicities for sl(r + 1) and the spectrum of the exterior algebra of the adjoint representation, J. Algebraic Combin. 1 (1992), 7-22.
[10] C. Carree, The rule of Littlewood-Richardson in a construction of Berenstein-Zelevinsky, Internat. J. Algebra Comput. 1 (1991), 473-491.
[11] J.B. Carrell, Chern classes of the Grassmannians and the Schubert calculus,Topology 17 (1978), 177-182.
[12] A.L. Cauchy, Memoire sur les fonctions qui ne peuvent otenir que deux valeurs egales et de signes contraires par suite des transpositions operes entre les variables qu'elles renferment, J. Ecole Polyt. 10 (1815), 29-112; Oeuvres,ser.2, vol.1, 91-169.
[13] W.Y.C. Chen, Log-concavity and q-log-convexity conjectures on the longest increasing subsequences of permutations, arXiv:0806.3392v2.
[14] W.Y.C. Chen, L.X.W. Wang, and A.L.B. Yang, Schur positivity and the q-log-convexity of the Narayana polynomials, arXiv:0806.1561v1.
[15] W.Y.C. Chen, L.X.W. Wang, and A.L.B. Yang, Recurrence relations for strongly q-log-convex polynomials, Canad. Math. Bull., to appear.
[16] T. Doslic and D. Veljan, Calculus proofs of some combinatorial inequalities,Math. Inequal. Appl. 6 (2003), 197-209.
[17] S. Fomin and C. Greene, A Littlewood-Richardson miscellany, European J.Combin. 14 (1993), 191-212.
[18] W. Fulton and J. Harris, Representation Theory, GTM, Vol. 129, Springer,Berlin, 1991.
[19] W. Fulton, Young Tableaux, Cambridge University Press, 1997.
[20] I.M. Gelfand and M.L. Tsetlin, Finite-dimensional representations of the group of unimodular matrices, Dokl. Akad. Nauk. SSSR 77 (1950), 825-828.
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