ANALYSIS OF STRAIGHTENING FORMULA

. The straightening formula has been an essential part of a proof showing that the set of standard bitableaux (or the set of standard monomials in minors) gives a free basis for a polynomial ring in a matrix of indeterminates over a field .The straightening formula expresses a nonstandard bitableau as an integral linear cobmbination of standard bitableaux. In this paper we analyse the exchanges in the process of straightening a nonstandard pure tableau of depth two.We give precisely the number of steps required to straighten a given violation of a nonstandard tableau.We also characterise the violation which is eliminated in a single step.


INTRODUCTION.
The straightening formula has been an intregral part of the theorem showing that the set of all standard monomials in minors of a matrix of indeterminates form a free basis of the polynomial ring in those indeterminates. It tranforms a nonstandard bitableau into an integral linear combination of the standard bitableaux, which makes sense only after using a correspondence between bitableaux and monomials in minors. The straightening formula is given in Rota-Doubillet -Stein [1] first and given again and exploited greatly in Desarmenien-Kung-Rota [2], DeConcini-Procesi [3], DeConcini-Eisenbud Procesi [4]. Abhyankar [5] gives a proof of the above mentioned theorem by explicitly counting tile dimension of a vector space generated by all the standard bitableaux of area V and length less than or equal to p, and deduces the result that the ideal generated by the p by p minors of a matrix of indeterminates is Hilbertian. In this exposition one also finds a form of the straightening formula which is very amenable for analysis. In a sequel of [5], Abhyankar has proved the Hilbertianness of a much more general determinantal ideal following the same strategy of counting and straightening, eliminating the proof of linear independence of standard bitableaux [6]. He also states the straightening formula in much general form and proposes the Problem: Given a nonstandard bitableau T, if T Y. c 1T is the expression for T given by the Straightening formula where T i's are standard bitableaux; can one determine c/s in terms of T ?
He defines the final integer function there which helps to give the coefficients ct. He also gives a recursion satisfied by this fin function, and states a problem of finding fin in terms of a given nonstandard bitableau. More knowledge about the number of steps required to straighten a given nonstandard bitableau will help finding this fin function.
In this paper we analyse the formula as given in [5]. For an analysis of the straightening formula for a nonstandard unitableau it is enough to look at a nonstandard pure unitableau of depth two. We state a form of the straightening formula using an arbitrary violation. The proof of this form is identical to the proof in [5]. We give an exact number of steps required to eliminate the violation from all unitableaux obtained in the straightening. As a part of our proof we give a detailed analysis of exchanges in the straightening and characterise a violation which gets eliminated in a single step (a good violation ).

NOTATION AND TERMINOLOGY.
Let Xij ]l<_ <_m,1 _< _<n be a matrix all of whose entries are indeterminates over a field K. Let Y be an m by m+n matrix formed by keeping the first m columns of Y to be those of X and putting the (n+i)th column to be (m-i+ 1)th column of an m by m identity matrix for < < m. Throughout the discussion we use the word "minor with the meaning as "determinant of a minor". In the proof of a theorem, slowing the set of standard monomials in maximal size minors of Y to be a free basis of K[Y], the spanning part of it is done by repeated applications of the straightening formula to a nonstandard unitableau of pure length m and bounded by m+n. Using this, one proves that the standard monomials in minors of X form a free basis of K[X] by invoking the correspondence between all minors of X and the maximal size minors of Y, and then the correspondence between tableaux and monomials in minors of X.
A univector of length m and bounded by p is an increasing sequence of m positive integers which are bounded by p. To a univector of length m and bounded by p there corresponds an m by m a maximal'size) mirror of an m by p matrix of indeterminates which is formed by picking up the corresponding m columns. A unitableau of depth d is a sequence of d univectors, written as A(1)A(2) A(d). By a pure unitableau of length m and bounded by p we mean a unitabeau each constituent univector has length m and is bounded by p. Given two univectors A= (A 1< A 2 <...<Ap)andB= (B < B 2 < <Bq ), we say that A<B ifp >q andA <B for <i <q. (2.1) A monomial in maximal size minors of an m by p matrix of indeterminates corresponding to a standard pure unitableau of length m and bounded by p is said to be standard.
For analysis one has to concentrate on unitableaux of pure length m and depth two. Let all unitableaux be bounded by p hereonwards. Let morn (X, AB be a monomial in maximal size minors of an m by p matrix X of indeterminates over a field K. Given a unitableau AB of pure length m and depth two as   PROOF. The proof of this theorem is the same as that in [5]  PROOF: Since every element of { A1,A2 Avq-} U b is smaller than every element of { A v q,Av q + A m }\ a, and by (4.0) and the given, card({A1,A 2 Av_q_ 1}Ub )=v-q-l+r >v, Bm}Ua,andby(4.0), From the above Theorem it is clear that starting with AB and applying the straightening formula N AB v times we can express AB or mom( X, AB as an integral linear combination of unitableaux which do not have v in their violation sets. In this process we do not perform any cancellations as we go. With this in mind, talking of the number of steps required to straighten a nonstandard unitableau is not confusing. By the above theorem and noting that the oddity function drops at each step, it follows that a nonstandard unitableau AB can be straightened in at most 1 < <m N AB (i) steps.

ACKNOWLEDGEMENTS.
would like to thank Professor Shreeram S. Abhyankar for his continuous encouragement and guidance while doing my Ph. D. thesis with him. would like to thank the Bhaskaracharya Pratishthana for its hospitality.