Enumerative combinatorics of Young tableaux.

*(English)*Zbl 0643.05001A unitableau \((p_{ij})\) is a tabular arrangement of positive integers \(p_{ij}\) taken from some set \(\{\) 1,2,...,p\(\}\), with strictly increasing rows. Unitableau is said to be standard (strongly standard) if its row lengths are nonincreasing and its columns are (strictly) increasing also. Strongly standard unitableaux were introduced by A. Young (1901) in his work on invariant theory. Thereafter he used them for describing the irreducible representations of the symmetric group; Young was led to this theme while investigating how Gordan-Capelli series, occurring in the classical invariant theory of forms, can be derived from an identity involving Young tableaux. Since then, Young tableaux have played an important role in quantum mechanics (H. Weyl, 1930-1940), they have occurred in the theory of elementary particles, in many combinatorial and computer science problems. W. Hodge (1947) used Young tableaux to study Schubert varieties of flag manifolds.

This monograph is a detailed and carefully commented research account, resulting from the author’s interest (1982-1985) in the structure of Schubert varieties. Having investigated certain determinantal ideals, he was led to the problem of enumerating Young tableaux and especially some related, more general objects - multitableaux.

To give idea of the results in the book, some definitions are needed. A tableau with q sides of the same shape but with possibly different entries, is called a multitableau (or simply, a tableau) of width q; in the special case \(q=2\) it is called bitableau. The tableau of width q having only one row, is called a multivector of width q. A tableau is said to be bounded by \(m,m=(m_ 1,...,m_ q)\in {\mathbb{N}}^ q\), if for any \(j\in \{1,...,q\}\) all the entries on the j-th side are \(\leq m_ j\). A few words more about these bitableaux. Let \(X=(x_{ij})\) be an \(m_ 1\times m_ 2\)-matrix with its elements being indeterminates over a field K. A bitableau bounded by \(m=(m_ 1,m_ 2)\) is a sequence of bivectors bounded by m, and each such bivector indicates some minor of X. Taking the product of all these minors for a given bitableau, we get a monomial in the minors of X; this monomial is called standard if the given tableau is standard. The Straightening Formula [see J. Désarménien, J. P. Kung and G.-C. Rota, Adv. Math. 27, 63-92 (1978; Zbl 0373.05010)] says, that the standard monomials of X form a K-basis of the algebra K[X] of polynomials in \(x_{ij}\). Some more definitions. The number of entries (rows) on each side of the tableau is called its area (depth), the length of the tableau is its largest row length. A standard tableau of width q is said to be predominated by a multivector a of width q if a is bounded by m and the tableau, is again standard.

Chapter 1 contains comments about new and complicated notation, some preliminary remarks, and a systematic and extensive treatment of binomial coefficients. Chapter 2 gives formulas for counting the sets stab(q,T) and \(mon(2,T)\), where \(stab(q,T)\) is the set of all standard tableaux of width q and area V, which are bounded by m and predominated by the multivector a of width q and length p, and \(mon(q,T)\) is the corresponding set of monomials. These formulas are examples of determinantal polynomials in binomial coefficients. Chapter 3 contains a certain universal identity, satisfied by minors of X. Chapter 4 gives several applications of these results: enumerative proofs of the Straightening Formula and of certain generalizations of the Second Fundamental Theorem of invariant theory, computations of Hilbert functions of determinantal ideals in K[X].

All chapters, though parts of the whole, are self-contained. They begin with an informal discussion, contain a summary, and motivation and hints about further use of the main results, as well as comments on underlying principles. Some useful illustrations and mental experiments are given for better understanding of basic points of reasoning. Proofs are divided into a sequence of independent statements, all these steps are numbered and their interrelations are indicated. The new symbol-codes, regardless of being quite puzzling locally, are suitably in the whole and form a complete system.

This book will be a valuable reference for research and applications of enumerating multitableaux, it may be used also as a text in the present- day-combinatorics for graduate courses.

This monograph is a detailed and carefully commented research account, resulting from the author’s interest (1982-1985) in the structure of Schubert varieties. Having investigated certain determinantal ideals, he was led to the problem of enumerating Young tableaux and especially some related, more general objects - multitableaux.

To give idea of the results in the book, some definitions are needed. A tableau with q sides of the same shape but with possibly different entries, is called a multitableau (or simply, a tableau) of width q; in the special case \(q=2\) it is called bitableau. The tableau of width q having only one row, is called a multivector of width q. A tableau is said to be bounded by \(m,m=(m_ 1,...,m_ q)\in {\mathbb{N}}^ q\), if for any \(j\in \{1,...,q\}\) all the entries on the j-th side are \(\leq m_ j\). A few words more about these bitableaux. Let \(X=(x_{ij})\) be an \(m_ 1\times m_ 2\)-matrix with its elements being indeterminates over a field K. A bitableau bounded by \(m=(m_ 1,m_ 2)\) is a sequence of bivectors bounded by m, and each such bivector indicates some minor of X. Taking the product of all these minors for a given bitableau, we get a monomial in the minors of X; this monomial is called standard if the given tableau is standard. The Straightening Formula [see J. Désarménien, J. P. Kung and G.-C. Rota, Adv. Math. 27, 63-92 (1978; Zbl 0373.05010)] says, that the standard monomials of X form a K-basis of the algebra K[X] of polynomials in \(x_{ij}\). Some more definitions. The number of entries (rows) on each side of the tableau is called its area (depth), the length of the tableau is its largest row length. A standard tableau of width q is said to be predominated by a multivector a of width q if a is bounded by m and the tableau, is again standard.

Chapter 1 contains comments about new and complicated notation, some preliminary remarks, and a systematic and extensive treatment of binomial coefficients. Chapter 2 gives formulas for counting the sets stab(q,T) and \(mon(2,T)\), where \(stab(q,T)\) is the set of all standard tableaux of width q and area V, which are bounded by m and predominated by the multivector a of width q and length p, and \(mon(q,T)\) is the corresponding set of monomials. These formulas are examples of determinantal polynomials in binomial coefficients. Chapter 3 contains a certain universal identity, satisfied by minors of X. Chapter 4 gives several applications of these results: enumerative proofs of the Straightening Formula and of certain generalizations of the Second Fundamental Theorem of invariant theory, computations of Hilbert functions of determinantal ideals in K[X].

All chapters, though parts of the whole, are self-contained. They begin with an informal discussion, contain a summary, and motivation and hints about further use of the main results, as well as comments on underlying principles. Some useful illustrations and mental experiments are given for better understanding of basic points of reasoning. Proofs are divided into a sequence of independent statements, all these steps are numbered and their interrelations are indicated. The new symbol-codes, regardless of being quite puzzling locally, are suitably in the whole and form a complete system.

This book will be a valuable reference for research and applications of enumerating multitableaux, it may be used also as a text in the present- day-combinatorics for graduate courses.

Reviewer: U.Kaljulaid

##### MSC:

05-02 | Research exposition (monographs, survey articles) pertaining to combinatorics |

05A15 | Exact enumeration problems, generating functions |

14M15 | Grassmannians, Schubert varieties, flag manifolds |

14M12 | Determinantal varieties |

05A10 | Factorials, binomial coefficients, combinatorial functions |

13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |