Topics in Hyperplane arrangements - Errata
aa r X i v : . [ m a t h . C O ] J un TOPICS IN HYPERPLANE ARRANGEMENTS, POLYTOPES, AND BOXSPLINES–ERRATA
C. DE CONCINI, C. PROCESI
Abstract.
We have received an e-mail from Bryan Gillespie pointing out that a proposition, thatis Proposition 8.5 of our book [1], is incorrect as stated. The given formula (8.5) is valid only inthe generic case that is assuming that for any point of the arrangement p , X p is formed by a basis.The correct proposition is slightly weaker, in general one must replace Formula 8.5 of the bookwith the next Formula (3). Accordingly one has to change Proposition 9.2 in the obvious way. Theremaining parts of the book are not affected but one should remove the first line of 11.3.3 whichquotes the incorrect formula. Here we discuss the correct proposition, replacing Proposition 8.5. Let us first develop a simple identity. Take vectors b i , i = 0 , . . . , k . Assume that b = P ki =1 α i b i . Choose numbers ν i , i = 0 , . . . , k, and set(1) ν := ν − k X i =1 α i ν i . If ν = 0 , we write(2) 1 Q ki =0 ( b i + ν i ) = ν − b + ν − P ki =1 α i ( b i + ν i ) Q ki =0 ( b i + ν i ) . When we develop the right-hand side, we obtain a sum of k + 1 terms in each of which one of theelements b i + ν i has disappeared. Let us remark that if α i = 0 the span of b . . . , ˇ b i , . . . b k equalsthe span of b , . . . , b k .Let us recall some notation, we let X = { a , . . . , a m } be a list of vectors spanning a real (or complex)vector space V and µ := { µ , . . . , µ m } a list of real (resp. complex) parameters. These data definea hyperplane arrangement in V ∗ given by the linear equations a i + µ i = 0, the various intersectionsof these hyperplanes form the subspaces of the arrangement. In particular we have the points ofthe arrangement for which we use the notation P ( X, µ ) of Section 2.1.1. Given p ∈ P ( X, µ ) we set X p for the sublist of a ∈ X such that a + µ a vanishes at p . Denote by L X p the family of subsets of X p spanning V . Notice that if ℓ ∈ L X p the linear polynomials a + µ a with a ∈ ℓ have p as uniquecommon zero. Proposition 1 (Replaces 8.5 of [1]) . Assume that X spans V . Then: (3) Y a ∈ X a + µ a = X p ∈ P ( X,µ ) X ℓ ∈L ( X p ) c ℓ Y a ∈ ℓ a + µ a = X p ∈ P ( X,µ ) X ℓ ∈L ( X p ) c ℓ Y a ∈ ℓ a − h a | p i with c ℓ ∈ C .For any p ∈ P ( X, µ ) , c X p = Y a ∈ X \ X p h a | p i + µ a . Proof.
This follows by induction applying the previous algorithm of separation of denominators.Precisely, if X is a basis, there is a unique point of the arrangement and there is nothing to prove.Otherwise, we can write X = ( Y, z ) where Y still spans V . By induction Y a ∈ X a + µ a = 1 z + µ z Y a ∈ Y a + µ a = X p ∈ P ( Y,µ ) X ℓ ∈L ( Y p ) c ℓ z + µ z Y a ∈ ℓ a + µ a . We need to analyze each product(4) 1 z + µ z Y a ∈ ℓ a + µ a . If h z | p i + µ z = 0 , then p ∈ P ( X, µ ) , ℓ ∪ { z } ∈ L ( X p ) and we are done. Otherwise,since ℓ spans V , write z = P a ∈ ℓ d a a and apply the previous algorithm to the list { z } ∪ ℓ and the correspondingnumbers µ z , µ a . As we have remarked the product (4) develops as a linear combination of productsof the form Y a ∈ ℓ ′ a + µ a ℓ ′ a proper subsequence of { z } ∪ ℓ and hence of X whose elements span V . So we can proceed byinduction.It remains to compute c X p . For a given p ∈ P ( X, µ ), Y a ∈ X \ X p a + µ a = c X p + X q ∈ P ( X,µ ) X ℓ ∈L ( X q ) ,ℓ = X p c ℓ Q a ∈ X p ( a + µ a ) Q a ∈ ℓ ( a + µ a ) . Hence, evaluating both sides at p yields c X p = Y a ∈ X \ X p h a | p i + µ a . (cid:3) Given any list X spanning V it is easily seen that for generic values of the parameters each set X p is a basis of V extracted from X and each basis of V extracted from X gives rise to a point of thearrangement. We say then that X, µ are generic.
Remark 1.
In case the data
X, µ are generic the set L ( X p ) reduces to the single element X p andFormula (3) gives back Formula 8.5. One can also reformulate the formula as(5) Y a ∈ X a + µ a = X p ∈ P ( X,µ ) C p Y a + µ a ∈ X p a + µ a with C p no more a number but a polynomial. OPICS IN HYPERPLANE ARRANGEMENTS, POLYTOPES, AND BOX SPLINES–ERRATA 3
In fact we can replace each term Y a ∈ ℓ a + µ a = Y a ∈ X p \ ℓ ( a + µ a ) Y a ∈ X p a + µ a and then collect the terms so that(6) C p = X ℓ ∈L ( X p ) c ℓ Y a ∈ X p \ ℓ ( a + µ a ) . As the reader will notice, if ℓ = X p , there is no Formula for the coefficients c ℓ this is due to thefact that these coefficients are not uniquely determined, that is the expansion of Formula (3) is ingeneral not unique, which is clear from Formula (6). References [1] C. De Concini and C. Procesi.