The Normal Form of Compound and Induced Matrices

Aitken, A. C.

doi:10.1112/plms/s2-38.1.354

Cited by 27 publications

(25 citation statements)

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“…193-203] claims to prove that L(m, n) is indeed a symmetric chain order for all m and n. However, his proof is invalid. Specifically, it relies on the "method of chains" of Aitken [45], and this method is not correct as stated by Aitken. For the reader's benefit we will discuss the nature of Aitken's error in more detail. Let P {X1, Xnt be a finite poset, and let (a) be the n x n matrix defined by a 0 unless x < x in P; otherwise the ai's are independent indeterminates over .Remove a chain C1 of maximum cardinality c from P, then remove a chain C2 of maximum cardinality c2 from P-C, etc.…”

mentioning

confidence: 99%

Weyl Groups, the Hard Lefschetz Theorem, and the Sperner Property

Stanley¹

1980

SIAM. J. on Algebraic and Discrete Methods

357

292

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Abstract. Techniques from algebraic geometry, in particular the hard Lefschetz theorem, are used to show that certain finite partially ordered sets O x derived from a class of algebraic varieties X have the k-Sperner property for all k. This in effect means that there is a simple description of the cardinality of the largest subset of C) x containing no (k + 1)-element chain. We analyze, in some detail, the case when X G/P, where G is a complex semisimple algebraic group and P is a parabolic subgroup. In this case, Qx is defined in terms of the "Bruhat order" of the Weyl group of G. In particular, taking P to be a certain maximal parabolic subgroup of G SO(2n + 1), we deduce the following conjecture of Erd6s and Moser: Let S be a set of 2 + 1 distinct real numbers, and let T1, , Tk be subsets of S whose element sums are all equal. Then k does not exceed the middle coefficient of the polynomial 2(1 + q)2(1 + q2)2... (1 + qe)2, and this bound is best possible.1. The Sperner property. Let P be a finite partially ordered set (or poser, for short), and assume that every maximal chain of P has length n. We say that P is graded of rank n. Thus P has a unique rank function p:P-{0, 1,..., n} satisfying p(x)= 0 if x is a minimal element of P, and p(y) p(x) + 1 if y covers x in P (i.e., if y > x and no z 6 P satisfies y > z > x). If p (x) i, then we say that x has rank i. Define Pi {x P: p (x) i} and set pi pi(P) card Pi. The polynomial F(P, q) po + plq +" + Pnq is called the rank-generating function of P. We say that P is rank-symmetric if pi pn-for all i, and that P is rank-unimodal if po <= pl <=" <= pi >= p+ >=" >= pn for some i.An antichain (also called a Spernerfamily or clutter) is a subset A of P, such that.no two distinct elements of A are comparable. The poset P is said to have the Sperner property (or property $1) if the largest size of an antichain is equal to max {pi: 0 <= <-n}.More generally, if k is a positive integer then P is said to have the k-Sperner property (or property Sk) if the largest subset of P containing no (k + 1)-element chain has cardinality max {PI +" "+ Pik 0 <= i <. < ik <= n}. If P has property S for all k =< n, then following [21] we say that P has property S. For further information concerning the Sperner property and related concepts, see for instance [15], [16], [17].Using some results from algebraic geometry, we will give several new classes of graded posets which have property S. These posets will all be rank-symmetric and rank-unimodal. First we must consider a property of posets related to property S.Suppose P is graded of rank n and is rank-symmetric. Again following [21], we say that P has property T if for all 0 -< i-< [n/2], there exist p pairwise disjoint saturated chains xi < xi+a <" < xn-i where xj P.. It is clear that P is then rank-unimodal. LEMMA 1.1. Let P be a finite graded rank-symmetric poset of rank n. The following three conditions are equivalent:(i) P is rank-unimodal and has property S.(ii) P has property T.

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confidence: 99%

Weyl Groups, the Hard Lefschetz Theorem, and the Sperner Property

Stanley¹

1980

SIAM. J. on Algebraic and Discrete Methods

357

292

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“…/˝J m . / was enunciated at that time (see [1,12,16]). However, it was not until rather later that a correct proof of this result ( [4,13]) was given.…”

Section: Tensor Products Of Jordan Matricesmentioning

confidence: 95%

Weitzenböck derivations of nilpotency 3

Wehlau

2012

Forum Mathematicum

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We consider a Weitzenböck derivation acting on a polynomial ring R D KOE 1 ; 2 ; : : : ; m over a field K of characteristic 0. The K-algebra R D ¹h 2 R j .h/ D 0º is called the algebra of constants. Nowicki considered the case where the Jordan matrix for acting on R 1 , the degree 1 component of R, has only Jordan blocks of size 2. He conjectured that a certain set generates R in that case. Recently Khoury, Drensky and Makar-Limanov and Kuroda have given proofs of Nowicki's conjecture. Here we consider the case where the Jordan matrix for acting on R 1 has only Jordan blocks of size at most 3. We use combinatorial methods to give a minimal set of generators G for the algebra of constants R . Moreover, we show how our proof yields an algorithm to express any h 2 R as a polynomial in the elements of G . In particular, our solution shows how the classical techniques of polarization and restitution may be used to augment the techniques of SAGBI bases to construct generating sets for subalgebras.

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“…Results concerning the Jordan form of tensor products were first obtained in the 1930s by Aitkin, Roth, and Littlewood [1,8,6]. Some gaps in the original proofs were filled in more recently by Marcus and Robinson [7] and by Brualdi [3].…”

Section: Introductionmentioning

confidence: 94%