Class Note for MATH 796 at KU
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This 2 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Kansas taught by a professor in Fall. Since its upload, it has received 15 views.
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Date Created: 02/06/15
Monday 324 The Big Face Lattice Let A A be the following two af ne line arrangements in R Are they isomorphic They have the same intersection poset and therefore the same characteristic polynomial which happens to be k2 7 516 6 but nonisomorphic dual graphs7only the dual graph of A has a vertex of degree 4 Therefore weld like to have some notion of isomorphism of real hyperplane arrangements that distinguishes between these two De nition 1 Let A H1 i i i Hn C R be a hyperplane arrangement and let 1 i i i Zn be linear forms such that Hi E 6 Rd l 0 Let c chi i i cn where Ci 6 70 for each if Consider the system of equations and inequalities gt 0 ifci lt 0 ifci 7 0 if c 0 If the solution set of this system is nonempty it is called a face of A and c is called a covectori The set of all faces is denoted Example Let A 2 Let H1 and H2 be the z and yaxes respectively so that we may take 41 y y and 21 y If The members of yQA are as follows Name Set Covector Origin 0 0 00 Positive zaxis 1 0 l I gt 0 0 Negative zaxis 1 0 l I lt 0 70 Positive y axis 0y l y gt 0 0 Negative y axis 0y l y lt 0 07 1st quadrant l I gt 0y gt 0 2nd quadrant l I lt 0y gt 0 7 3rd quadrant l I lt 0y lt 0 7 4th quadrant l I gt 0y lt 0 7 The set yQA has a natural partial ordering given by F S F whenever F E F where the bar denotes closure in the usual topology on Rail Equivalently if 50 are the covectors of F F respectively then Ci 6 020 for every ii Proposition 1 The partially ordered set yQA U 0l is a ranked lattice called the big face lattice of A Note The adjective big modi es lattice not face For example the Hasse diagram of 232 is shown on the left of the gure belowi Since the Hasse diagram can be quite messy it is typically more useful to draw a picture of A in Which each face is labeled by its covector as on the right l 0 7 H 7 7 0 0 0 0 70 O 0 W 00 00 quot 0 0 If F is a face of A With covector c then the af ne span of F of A is an intersection of hyperplanes in A namely those for Which Ci 0 Therefore we can recover the intersection poset LA from The coatoms of are the regions of Rd A The corresponding maximal covectors consist entirely of 7s and 77s With no Us We can recover the dual graph of A from yA because two maximal covectors represent adjacent regions if and only if they differ in exactly one digiti
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