On the Generic Point Arrangements in Euclidean Space and Stratification of the Totally Nonzero Grassmannian
aa r X i v : . [ m a t h . G M ] F e b ON THE GENERIC POINT ARRANGEMENTS IN EUCLIDEAN SPACEAND STRATIFICATION OF THE TOTALLY NONZEROGRASSMANNIAN
AUTHOR: C.P. ANIL KUMAR*A bstract . In this article, for positive integers n ≥ m ≥
1, the parameter spacesfor the isomorphism classes of the generic point arrangements of cardinality n ,and the antipodal point arrangements of cardinality 2 n in the Eulidean space R m are described using the space of totally nonzero Grassmannian Gr tnzmn ( R ) .A stratification S tnzmn ( R ) of the totally nonzero Grassmannian Gr tnzmn ( R ) is men-tioned and the parameter spaces are respectively expressed as quotients of thespace S tnzmn ( R ) of strata under suitable actions of the symmetric group S n and thesemidirect product group ( R ∗ ) n ⋊ S n . The cardinalities of the space S tnzmn ( R ) ofstrata and of the parameter spaces S n \S tnzmn ( R ) , (( R ∗ ) n ⋊ S n ) \S tnzmn ( R ) are enu-merated in dimension m =
2. Interestingly enough, the enumerated value of theisomorphism classes of the generic point arrangements in the Euclidean planeis expressed in terms of the number theoretic Euler-totient function. The analo-gous enumeration questions are still open in higher dimensions for m ≥ . Introduction
The stratification of Grassmannians Gr mn ( R ) is an interesting topic of study.There are various types of stratifications of different subsets of Grassmannians.A stratification is a decomposition of the Grassmannian or a subset of the Grass-mannian into various strata where each stratum satisfies some nice properties.Some of the decompositions that exist in the literature are:( ) the decomposition of the Grassmannian Gr mn ( R ) into Schubert cells { Ω λ ( R ) | λ ⊆ ( n − m ) m } indexed by partitions λ ⊆ ( n − m ) m ,( ) the decomposition of the Grassmannian Gr mn ( R ) into matroid strata {S M ( R ) | M a realizable matroid of rank m } also known as Gelfand-Serganova strata labelled by some matroids M called realizable matroids,( ) the decomposition of the totally nonnegative Grassmannian Gr tnzmn ( R ) intototally nonnegative Grassmann cells {S tnn M ( R ) = S M ( R ) ∩ Gr tnnmn ( R ) | Date : F ebruary 23 , . Mathematics Subject Classification.
Primary: M . Key words and phrases.
Grassmannians, Projective Spaces, Point Arrangements.*The work is done while the author is a Post Doctoral Fellow at Harish-Chandra ResearchInstitute, Allahabad. C.P. ANIL KUMAR M a realizable matroid of rank m such that S M ( R ) ∩ Gr tnnmn ( R ) = ∅ } ob-tained from the matroid strata S M ( R ) for various M .Moreover in cases ( ) and ( ), the topological structure of a cell/stratum is home-omorphic to an open ball of appropriate dimension, that is, they are actuallycells. In case ( ), Ω λ ( R ) ∼ = R | λ | . In case ( ), S tnn M ∼ = ( R + ) d for some suitable d ≥
0. In case ( ), the geometric structure of a matroid stratum S M ( R ) can behighly nontrivial as N. E. Mn¨ev [ ] has shown that, it can be as complicated asessentially any algebraic variety.The enumeration of the number of Schubert cells in the decomposition of Gr mn ( R ) in case ( ) is ( nm ) a binomial coefficient and the enumeration of the totally non-negative Grassmann cells in case ( ) is related to Eulerian numbers (refer Section in A. Postnikov [ ] and also L. K. Williams [ ]).In this article, we describe the stratification S tnzmn ( R ) of another subset of theGrassmannian namely, the totally nonzero Grassmannian Gr tnzmn ( R ) . We alsoenumerate the number of strata in S tnz n ( R ) , that is, for m = n − ( n − ) ! for n ≥
2. The enumeration question is still open for dimensions m >
2. Also we relate the totally nonzero Grassmannian Gr tnzmn ( R ) to the genericpoint arrangements of cardinality n and the antipodal point arrangements ofcardinality 2 n in Euclidean space R m and describe the parameter spaces for theisomorphism classes of the generic point arrangements and the antipodal pointarrangements via certain group actions on the space S tnzmn ( R ) of strata. Again weenumerate the parameter spaces S n \S tnz n ( R ) , (( R ∗ ) n ⋊ S n ) \S tnz n ( R ) for m = S n \S tnz n ( R ) has an interesting answer.The definitions and details are mentioned in the next and later sections. . Point Arrangements in Euclidean SpacesDefinition . (A Point Arrangement, An Generic Point Arrangement, An An-tipodal Point Arrangement) . Let n ≥ m ≥ S ⊂ R m is a said to be a point arrangement. A point arrange-ment S = { v , v , · · · , v n } ⊂ R m is said to be generic if any subset I ⊂ S of cardinality at most m is a linearly independent set. A point arrangement S = { v , v , · · · , v n } ⊂ R m is said to be an antipodal point arrangement if forevery 1 ≤ i ≤ n there exists unique 1 ≤ j = σ ( i ) ≤ n such that v i = tv j forsome real number t < I of cardinality n ≥ m consisting of onerepresentative from each of the sets { v i , v σ ( i ) } is a generic point arrangement in R m . Conventionally, we choose v σ ( i ) = − v i for every 1 ≤ i ≤ n and we call thesubset { v i , v σ ( i ) } a set consisting of an antipodal pair. OINT ARRANGEMENTS AND STRATIFICATION OF TOTALLY NONZERO GRASSMANNIAN Definition . (Isomorphic Generic Point Arrangements, Isomorphic AntipodalPoint Arrangements) . We say two generic point arrangements S , S in R m areisomorphic if there is a set bijection σ : S −→ S such that for any subset C = { v i , v i , · · · , v i m , v i m + } ⊆ S of cardinality ( m + ) if v i m + = m ∑ j = a j v i j andif σ ( v i m + ) = m ∑ j = b j σ ( v i j ) then for each 1 ≤ j ≤ m , a j b j >
0, that is, sgn ( a j ) = sgn ( b j ) = S , S are isomorphic ifthere exists a bijection σ : S −→ S such that σ ( − v ) = − σ ( v ) for every v ∈ S and if for every subset C = { v i , v i , · · · , v i m , v i m + } ⊆ S of cardinality ( m + ) not containing any antipodal pair, if v i m + = m ∑ j = a j v i j and if σ ( v i m + ) = m ∑ j = b j σ ( v i j ) then a j > ≤ j ≤ m if and only if b j > ≤ j ≤ m . Remark . . Let S be a generic point arrangement in R m . Let A ∈ GL m ( R ) . Then T = AS = { Av | v ∈ S } is also a generic point arrangement in R m . Moreover S is isomorphic to T . Similarly if S is an antipodal point arrangement in R m then T is also an antipodal point arrangement in R m and S is isomorphic to T . . Grassmannians
For n ≥ m ≥
1, the Grassmannian Gr mn ( R ) is the collection of m -dimensionalsubspaces V ⊆ R n . It can be presented as the quotient Gr mn ( R ) = GL m ( R ) \ Mat ∗ mn ( R ) ,where Mat ∗ mn ( R ) is the space of ( m × n ) -matrices of rank m . Here we assumethat the subspace V associated with a ( m × n ) -matrix M is spanned by the rowvectors of M . . . Pl ¨ucker coordinates.
For a ( m × n ) -matrix M and a m -element subset I ⊆ [ n ] = {
1, 2, · · · , n } , let M I denote the ( m × m ) -submatrix of M in the column set I , and let ∆ I ( M ) : = Det ( M I ) denote the maximal minor of M . If we mul-tiply M by A ∈ GL m ( R ) on the left, all minors ∆ I ( M ) are rescaled by thesame factor Det ( A ) . If M = ( m ij ) is in I -echelon form then M I = Id m and m ij = + ∆ I \{ i }∪{ j } or m ij = − ∆ I \{ i }∪{ j } and the sign can be determined. Thusthe ( ∆ I ) I ∈ ( [ n ] m ) form projective coordinates of the Grassmannian Gr mn ( R ) calledthe Pl ¨ucker coordinates and the map M −→ ( ∆ I ) I ∈ ( [ n ] m ) induces the Pl ¨ucker em-bedding Gr mn ( R ) ֒ → RP ( nm ) − of the Grassmannian into the projective space.The image of the Grassmannian Gr mn ( R ) under the Pl ¨ucker embedding is the C.P. ANIL KUMAR algebraic subvariety in RP ( nm ) − given by the Grassmann-Pl ¨ucker relations: ∆ ( i , i , ··· , i m ) . ∆ ( j , j , ··· , j m ) = m ∑ s = ∆ ( j s , i , ··· , i m ) . ∆ ( j , j , ··· , j s − , i , j s + , j s + , ··· , j m ) ,for any i , i , · · · , i m , j , j , · · · , j m ∈ [ n ] . Here we assume that ∆ ( i , i , ··· , i m ) (labelledby an ordered sequence rather than a subset) equals to ∆ { i , i , ··· , i m } if i < i < · · · < i m and ∆ ( i , i , ··· , i m ) = ( − ) sign ( w ) ∆ ( i w ( ) , i w ( ) , ··· , i w ( m ) ) for all w ∈ S m . . . The Totally Nonzero Grassmannian.Definition . . Let us define the totally nonzero Grassmannian Gr tnzmn ( R ) ⊂ Gr mn ( R ) as the quotient Gr tnzmn ( R ) = GL m ( R ) \ Mat tnzmn ( R ) where Mat tnzmn ( R ) is theset of real ( m × n ) -matrices M of rank m with all maximal minors ∆ I ( M ) = M = ( v , v , · · · , v n ) where v i is the i th -column of the matrix M for 1 ≤ i ≤ n then the set S = { v i | ≤ i ≤ n } is a generic point arrangement. Moreover if N = ( w , w , · · · , w n ) is another matrix representing the same element V of theGrassmannian then there exists a matrix A ∈ GL m ( R ) such that AM = N . Sothe set T = { w , w , · · · , w n } = AS is a generic point arrangement isomorphic to S . Hence each element V = GL m ( R ) . M ∈ Gr mn ( R ) represents an isomorphismclass of a generic point arrangement. . . Stratification of the Totally Nonzero Grassmannian.Definition . . Let
C ⊆ ( [ n ] m ) be a certain collection of m -subsets of {
1, 2, · · · , n } .Define the stratum S tnz C ( R ) = { GL m ( R ) . M ∈ Gr tnzmn ( R ) | either ∆ I ( M ) > I ∈ C and ∆ I ( M ) < I / ∈ C or ∆ I ( M ) < I ∈ C and ∆ I ( M ) > I / ∈ C } . Note S tnz C ( R ) = S tnz ( [ n ] m ) \C ( R ) . Let S tnzmn ( R ) = {S tnz C ( R ) | C ⊆ ( [ n ] m ) , S tnz C ( R ) = ∅ } denote the collection of nonempty strata. Remark . . Using Pl ¨ucker relations, we can produce a collection C whose stra-tum S tnz C ( R ) is empty. Remark . . The description of the totally positive Grassmannian and the totallynonnegative Grassmannian and their stratifications are given in A. Postnikov [ ]. . . Some Group Actions on Strata.Definition . . We define the action of ( R ∗ ) n on S tnzmn ( R ) . Let S tnz C ( R ) be a non-empty stratum, that is, S tnz C ( R ) ∈ S tnzmn ( R ) for some C ∈ ( [ n ] m ) . Let ( t , t , · · · , t n ) ∈ ( R ∗ ) n and GL m ( R ) . M ∈ S tnz C ( R ) where M = ( v , v , · · · , v n ) with v i ∈ R m a col-umn vector of M for 1 ≤ i ≤ n . Define N = ( t v , t v , · · · , t n v n ) and definethe action ( t , t , · · · , t n ) • GL m ( R ) . M = GL m ( R ) . N . We observe that ∆ I ( N ) = OINT ARRANGEMENTS AND STRATIFICATION OF TOTALLY NONZERO GRASSMANNIAN (cid:18) ∏ i ∈ I t i (cid:19) ∆ I ( M ) . Now define the collection D = (cid:8) I ∈ ( [ n ] m ) such that ∆ I ( N ) > (cid:9) .Then the action of ( R ∗ ) n on the strata is given by ( t , t , · · · , t n ) • S tnz C ( R ) = ( t , t , · · · , t n ) . S tnz ( [ n ] m ) \C ( R ) = S tnz D ( R ) = S tnz ( [ n ] m ) \D ( R ) . Definition . . We define the action of ( R + ) n on S tnzmn ( R ) . This action is therestriction of the action of ( R ∗ ) n . Here in fact we observe that ( t , t , · · · , t n ) •S tnz C ( R ) = S tnz C ( R ) . Every stratum is a fixed point for this action. So on thestrata this action is trivial. Definition . . We define the action of S n on S tnzmn ( R ) . Let P be an ( n × n ) -permutation matrix in S n . Let S tnz C ( R ) be a non-empty stratum and GL m ( R ) . M ∈S tnz C ( R ) . Then define P • ( GL m ( R ) . M ) = GL m ( R ) . MP − . If the permutation ma-trix P is denoted by σ : [ n ] −→ [ n ] then for 1 ≤ i ≤ n , P − ( e ni ) t = ( e n σ − ( i ) ) t where e ni is the standard n -dimensional row vector which has entries 1 at the i th placeand zeroes elsewhere and t stands for transpose. If M = ( v , v , · · · , v n ) then MP − = ( v σ − ( ) , v σ − ( ) , · · · , v σ − ( n ) ) . This action gives rise to a well-defined ac-tion on the strata, that is, if GL m ( R ) . M , GL m ( R ) . N belong to the same stratum S tnz C ( R ) then GL m ( R ) . MP − , GL m ( R ) . NP − belong to the same stratum. So σ • S tnz C ( R ) = S tnz D ( R ) where GL m ( R ) . MP − ∈ S tnz D ( R ) . Definition . . We define the semi-direct product group ( R ∗ ) n ⋊ S n as follows.Let ( s , s , · · · , s n ) , ( t , t , · · · , t n ) ∈ ( R ∗ ) n and σ , τ ∈ S n . Define (( s , s , · · · , s n ) , τ ) . (( t , t , · · · , t n ) , σ ) = (( s σ ( ) t , s σ ( ) t , · · · , s σ ( n ) t n ) , τσ ) .Let S tnz C ( R ) be a non-empty stratum and GL m ( R ) . M ∈ S tnz C ( R ) . Now define theaction of ( R ∗ ) n ⋊ S n on the element GL m ( R ) . M ∈ Gr tnzmn ( R ) as (( t , t , · · · , t n ) , σ ) • ( GL m ( R ) . M ) = GL m ( R ) . N with N = ( t σ − ( ) v σ − ( ) , t σ − ( ) v σ − ( ) , · · · , t σ − ( n ) v σ − ( n ) ) where M = ( v , v , · · · , v n ) . This action gives rise to an action on the space S tnzmn ( R ) of strata as (( t , t , · · · , t n ) , σ ) • S tnz C ( R ) = S tnz D ( R ) where GL m ( R ) . N ∈ S tnz D ( R ) . Theorem . . Consider the action of S n on the space S tnzmn ( R ) of strata. Let C , D betwo collections of m-subsets of [ n ] such that S tnz C ( R ) , S tnz D ( R ) are non-empty. Then S tnz C ( R ) and S tnz D ( R ) are in the same orbit for the action of S n if and only if for anytwo matrices M = ( v , v , · · · , v n ) , N = ( w , w , · · · , w n ) ∈ Mat tnzmn ( R ) such that C.P. ANIL KUMAR GL m ( R ) . M ∈ S tnz C ( R ) , GL m ( R ) . N ∈ S tnz D ( R ) the sets S = { v , v , · · · , v n } andT = { w , w , · · · , w n } are isomorphic generic point arrangements.Proof. ( ⇐ ) Let S and T be two isomorphic arrangements and σ : T −→ S bean isomorphism. Let σ ( w i ) = v σ ( i ) , 1 ≤ i ≤ n . Let M = ( v , v , · · · , v n ) and N = ( w , w , · · · , w n ) P − = ( w σ − ( ) , w σ − ( ) , · · · , w σ − ( n ) ) where P is a permu-tation matrix such that P − ( e ni ) t = ( e n σ − ( i ) ) t . Then we prove that GL m ( R ) . N ∈S tnz C ( R ) which is the stratum that contains GL m ( R ) . M . First assume without lossof generality that v i = ( e mi ) t = w σ − ( i ) , 1 ≤ i ≤ n . Then Det ( v , v , · · · , v m ) = Det ( w σ − ( ) , w σ − ( ) , · · · , w σ − ( m ) ) =
1. Moreover for i > m we have if v i =( x i , x i , · · · , x im ) t and w σ − ( i ) = ( y i , y i , · · · , y im ) t then sgn ( x ij ) = sgn ( y ij ) for1 ≤ j ≤ m < i ≤ n . This follows because σ − : v i −→ w σ − ( i ) , 1 ≤ i ≤ n is an iso-morphism. Hence sgn ( Det ( v i , v i , · · · , v i m − , v i )) = sgn ( Det ( w σ − ( i ) , v σ − ( i ) , · · · , v σ − ( i m − ) , w σ − ( i ) )) for any 1 ≤ i < i < · · · < i m − ≤ m < i ≤ n . Sothe signs of the determinants agree for any m -subset I of [ n ] such that | I ∩ [ m ] | = m − m -subset [ m ] ⊆ [ n ] . Now the proof can be extended to all m subsets of [ n ] . Sothe elements GL m ( R ) . N , GL m ( R ) . M are in the same stratum. Hence the ele-ments GL m ( R ) . N , GL m ( R ) . M are in the same S n -orbit implying that the strata S tnz C ( R ) , S tnz D ( R ) are in the same S n -orbit.. ( ⇒ ) Now assume that the stratum S tnz C ( R ) , S tnz D ( R ) are in the same S n -orbit. Let σ • S tnz C ( R ) = S tnz D ( R ) . Let GL m ( R ) . M ∈ S tnz C ( R ) , GL m ( R ) . N ∈ S tnz D ( R ) where M = ( v , v , · · · , v n ) , N = ( w , w , · · · , w n ) , N σ − = ( w σ − ( ) , w σ − ( ) , · · · , w σ − ( n ) ) .If the signs of the coordinates ( ∆ I ( M )) I ∈ ( [ n ] m ) and ( ∆ I ( N σ − )) I ∈ ( [ n ] m ) agree uni-formly then we do not need to reflect N . Otherwise if the signs of the coordi-nates uniformly disagree then we reflect N by using a reflection R ∈ GL m ( R ) and consider RN which does not change the isomorphism class of N . Now theprevious proof can be traced back to obtain an isomorphism σ : T −→ S . So thegeneric point arrangements T and S are isomorphic. (cid:4) Theorem . . Consider the action of ( R ∗ ) n ⋊ S n the space S tnzmn ( R ) of strata. Let C , D be two collections of m-subsets of [ n ] such that S tnz C ( R ) , S tnz D ( R ) are non-empty.Then S tnz C ( R ) and S tnz D ( R ) are in the same orbit for the action of ( R ∗ ) n ⋊ S n ifand only if for any two matrices M = ( v , v , · · · , v n ) , N = ( w , w , · · · , w n ) ∈ Mat tnzmn ( R ) such that GL m ( R ) . M ∈ S tnz C ( R ) , GL m ( R ) . N ∈ S tnz D ( R ) the sets S = {± v , ± v , · · · , ± v n } and T = {± w , ± w , · · · , ± w n } are isomorphic antipodal pointarrangements.Proof. The proof is similar to the proof of Theorem . except here the groupwhich is acting on the strata is ( R ∗ ) n ⋊ S n . (cid:4) OINT ARRANGEMENTS AND STRATIFICATION OF TOTALLY NONZERO GRASSMANNIAN Theorem . . Let n ≥ m ≥ be two positive integers.( ) The isomorphism classes of the generic point arrangements of cardinality n inthe Euclidean space R m are in bijection with the orbits in S n \S tnzmn ( R ) under theaction of S n on the space S tnzmn ( R ) of strata.( ) The isomorphism classes of the antipodal point arrangements of cardinality n inthe Euclidean space R m are in bijection with the orbits in (( R ∗ ) n ⋊ S n ) \S tnzmn ( R ) under the action of ( R ∗ ) n ⋊ S n on the space S tnzmn ( R ) of strata.Proof. The theorem follows from Theorem . and Theorem . . (cid:4) . On the Strata in Dimension Two
First we represent any generic point arrangement in R combinatorially by cap-turing all the required geometric data combinatorially. . . Combinatorial Representation of a Generic Point Arrangement in R . Let S = { v , v , · · · , v n } be a generic point arrangement in R . Assume the Let v i =( x i , y i ) ∈ R . Note v i = ∈ R . Let the angle the line L i = { tv i | t ∈ R } makeswith respect to the positive X -axis be θ i . We note that the lines L i , 1 ≤ i ≤ n areall distinct. Assume after renumbering the subscripts of the elements of S wehave 0 ≤ θ < θ < · · · < θ n < π . A combinatorial representation of the set S is an element ( a , a , · · · , a n ) in the set { + − } × { + − } × · · · × { + n , − n } defined as follows. If y = a = sgn ( x ) else a = sgn ( y ) . We define for i > a i = sgn ( y i ) i . We note that for i > y i = Lemma . . If S = { v , v , · · · , v n } is a generic point arrangement and the line L i = { tv i | t ∈ R } makes an angle θ i with respect to the positive X-axis with ≤ θ < θ < · · · < θ n < π . Let ( a , a , · · · , a n ) ∈ n ∏ i = { + i , − i } be the combinatorial representationof the set S. Then we have for ≤ i = j ≤ n , sgn ( Det ( v i , v j )) = sgn ( a i a j ( j − i )) .Proof. The proof is an observation obtained by considering all the various cases. (cid:4)
Definition . . For a matrix M = ( v , v , · · · , v n ) ∈ Mat tnz n ( R ) we define theorientation sign matrix O M = [ sgn ( Det ( v i , v j ))] ≤ i , j ≤ n . Let O n ( R ) = { O M | M ∈ Mat tnz n ( R ) } be space of orientation sign matrices. Remark . . If M = ( v , v , · · · , v n ) ∈ Mat tnz n ( R ) and P is a permutation matrixassociated to the permutation σ : [ n ] −→ [ n ] then P − ( e ni ) t = ( e n σ − ( i ) ) t and forthe matrix N = ( v σ − ( ) , v σ − ( ) , · · · , v σ − ( n ) ) = MP − the orientation sign matrix O N = [ sgn ( Det ( v σ − ( i ) , v σ − ( j ) ))] ≤ i , j ≤ n = PO M P − (also denoted by σ O M σ − ). C.P. ANIL KUMAR
Remark . . If M = ( v , v , · · · , v n ) ∈ Mat tnz n ( R ) then the plucker coordinates ( ∆ I ( M )) I = { i , j }∈ ( [ n ] ) of GL ( R ) . M ∈ Gr tnz n ( R ) satisfy that its associated sign vectoris given either by the strictly upper triangular entries of the orientation signmatrix O M or it is given by the strictly lower triangular entries of O M . Moreoverwe note that if σ : [ n ] −→ [ n ] is the permutation which takes i to n − i +
1, 1 ≤ i ≤ n then − O M = O tM = σ O M σ − = O N where N = ( v n , v n − , · · · , v , v ) . Remark . . It is therefore easy to observe that the group Z Z = { + − } actson the set O n ( R ) of orientation sign matrices as − • O M = O N where M =( v , v , · · · , v n ) , N = ( v n , v n − , · · · , v , v ) ∈ Mat tnz n ( R ) and the set S tnz n ( R ) is inbijection with the orbit space Z Z \O n ( R ) . . . Combinatorial Representation of an Element in
Mat tnz n ( R ) . Now we ex-tend the definition of a combinatorial representation of a generic point arrange-ment to a combinatorial representation of an element in
Mat tnz n ( R ) . Let M =( v , v , · · · , v n ) ∈ Mat tnz n ( R ) . Let S = { v i | ≤ i ≤ n } . Let the combinatorialrepresentation of S be given by the element ( a , a , · · · , a n ) ∈ n ∏ i = { + i , − i } . Letthe line L i = { tv i | t ∈ R } make an angle θ τ ( i ) with respect to the positive X -axis for a permutation τ ∈ S n such that 0 ≤ θ < θ < · · · < θ n < π . Thenthe combinatorial representation of M is a permutation of the coordinates of theelement given by ( a τ ( ) , a τ ( ) , · · · , a τ ( n ) ) . This motivates the definition of signedpermutation group given in Definition . . . . Enumeration of Strata in Dimension Two.
In this section we enumeratethe number of strata in dimension two that is the cardinality of the set S tnz n ( R ) . Definition . . Let P n = { ( a , a , · · · , a n ) ∈ {± ± · · · , ± n } n | there exists π ∈ S n with π ( i ) = sgn ( a i ) a i } , be the signed permutation group. We have P n ∼ = S n ⋉ ( Z /2 Z ) n with the isomorphism being ( a , a , · · · , a n ) −→ (cid:0) π , ( sgn ( a ) , sgn ( a ) , · · · , sgn ( a n )) (cid:1) where π ( i ) = sgn ( a i ) a i . Let p = ( a , a , · · · , a n ) ∈ P n . The signed permutationmatrix associated to p is M = ( v , v , · · · , v n ) ∈ GL n ( R ) where v i = sgn ( a i ) e t π ( i ) and sgn ( a i ) a i = π ( i ) . The group multiplication in P n is given by the correspond-ing matrix multiplication as a subgroup of GL n ( R ) . Let K n ∼ = Z n Z be the cyclicsubgroup of order 2 n generated by the element ( − n , 1, · · · , ( n − ) , ( n − )) ∈ P n . Theorem . . ( ) The set { O M | M ∈ Mat tnz n ( R ) } of orientation sign matrices isin bijection with the rights cosets of K n in P n and its cardinality is n − ( n − ) ! for n ≥ . OINT ARRANGEMENTS AND STRATIFICATION OF TOTALLY NONZERO GRASSMANNIAN ( ) The set S tnz n ( R ) of strata in dimension two which is in bijection with the orbitspace Z Z \O n ( R ) has cardinality n − ( n − ) ! for n ≥ .Proof. Let M = ( v , v , · · · , v n ) ∈ Mat tnz n ( R ) be such that the combinatorialrepresentation of the generic point arrangement S = { v i | ≤ i ≤ n } be-longs to K n . Then there exists a permutation P σ ∈ S n such that MP − σ =( v σ − ( ) , v σ − ( ) , · · · , v σ − ( n )) and the orientation sign matrix O MP − σ = P σ O M P − σ is the standard matrix O n given by O MP − σ = O n = [ a ij ] ≤ i , j ≤ n with a ij = i < j , a ij = − i > j and a ij = i = j . Conversely it is also clear from the arrange-ment of points v i , 1 ≤ i ≤ n in the plane that, if the orientation matrix of O MP − σ is the standard matrix O n for some permutation P σ ∈ S n then the combinatorialrepresentation of S belongs to K n .Now we observe that the combinatorial representation itself, of M given by ( a , a , · · · , a n ) in P n , belongs to K n if and only if the orientation sign matrix O M is the standard matrix. So the combinatorial representation of M belongs to thecoset K n p − for p ∈ P n if and only if the orientation sign matrix O M = PO n P − where P is the associated signed permutation matrix of p and O n is the standardorientation sign matrix. Also any orientation sign matrix is a conjugate of O n by some element p ∈ P n . Hence ( ) follows and the number of right cosets is | P n || K n | = n n !2 n = n − ( n − ) !.Now ( ) follows since the orientation sign matrices O M and O tM = − O M = O RM = O N correspond to the same stratum using a reflection R of the plane.Here N = ( v n , v n − , · · · , v ) . Hence | Z Z \O n ( R ) | = n − ( n − ) !. This provesthe theorem. (cid:4) . . Enumeration of the Isomorphism Classes of the Antipodal Point Arrange-ments in the Plane.
The theorem is stated as follows.
Theorem . . The group (( R ∗ ) n ⋊ S n ) acts transitively on S tnz n ( R ) . Hence the orbitspace (( R ∗ ) n ⋊ S n ) \S tnz n ( R ) is a singleton set for n ≥ .Proof. It is clear geometrically that, if there are n -antipodal pairs in R formingan antipodal point arrangement in the plane then they can be arranged on lines L , L , · · · , L n passing through origin making angles θ i , 1 ≤ i ≤ n with respectto the positive X -axis such that 0 ≤ θ < θ < · · · < θ n < π . Hence thegroup (( R ∗ ) n ⋊ S n ) acts transitively on S tnz n ( R ) . Also group theoretically, wesee that the signed permutation group P n acts transitively on the set { O M | M ∈ Mat tnz n ( R ) } of orientation sign matrices by conjugation. Now the theoremfollows. (cid:4) C.P. ANIL KUMAR . . Enumeration of the Isomorphism Classes of the Generic Point Arrange-ments in the Plane.
In this section we enumerate the cardinality of the space oforbits S n \S tnz n ( R ) under the action of S n on the space of strata S tnz n ( R ) . Theorem . . Let n be a positive integer and ζ be a generator of the cyclic group Z n Z of order n. Consider the following action of Z n Z on the set ( Z Z ) n = { + − } n . Thegenerator ζ acts as: ζ • ( d , d , · · · , d n ) = ( − d n , d , d , · · · , d n − ) . Let ≤ i ≤ n − where n = l m ∈ N , 2 ∤ m , l ∈ N ∪ { } . Then | { d ∈ { + − } n | ζ i • d = d } | = n if i = gcd ( i , n ) if i = l + j ,0 otherwise . Proof.
Clearly the theorem holds for n =
1, 2.For any n ∈ N , if i = | { d ∈ { + − } n | ζ i • d = d } | = n .Let n = l m , 2 ∤ m ∈ N , l ∈ N ∪ { } . Let i = l + j , gcd ( j , m ) =
1. Suppose l = n = m ≥
3. Then ( d , d , · · · , d m ) ∈ { + − } m is a fixed point of ζ if andonly if ( − d m − , − d m , d , d , · · · , d m − ) = ( d , d , · · · , d m ) .So either d i = ( − ) i , 1 ≤ i ≤ m or d i = ( − ) i + , 1 ≤ i ≤ m . Hence there are onlytwo fixed points for ζ . Now ζ • d = d ⇒ ζ j • d = d and if j ′ ∈ {
1, 2, · · · , m − } such that jj ′ ≡ m then 2 jj ′ ≡ m and ζ j • d = d ⇒ ( ζ j ) j ′ • d = d ⇒ ζ • d = d .Hence we have | { d ∈ { + − } n | ζ j • d = d } | = | { d ∈ { + − } n | ζ • d = d } | = = gcd ( i , n ) .Let n = l m , 2 ∤ m ∈ N , l ∈ N ∪ { } . Let i = l + j , gcd ( j , m ) =
1. Suppose l ≥ n ′ = n , i ′ = i . So i = i ′ , n = n ′ . We have by induction, usingthe theorem for values smaller than l | { d ∈ { + − } n ′ | ζ i ′ • d = d } | = gcd ( i ′ , n ′ ) = l − .Let ( d , e , d , e , · · · , d n ′ , e n ′ ) ∈ { + − } n . We observe that ζ i ′ • ( d , e , d , e , · · · , d n ′ , e n ′ ) = ( d , e , d , e , · · · , d n ′ , e n ′ ) OINT ARRANGEMENTS AND STRATIFICATION OF TOTALLY NONZERO GRASSMANNIAN if and only if for < τ > = Z n ′ Z , τ • ( d , d , · · · , d n ′ ) = ( − d n ′ , d , · · · , d n ′ − ) wehave τ i ′ • ( d , d , · · · , d n ′ ) = ( d , d , · · · , d n ′ ) and τ i ′ • ( e , e , · · · , e n ′ ) = ( e , e , · · · , e n ′ ) .Hence | { d ∈ { + − } n | ζ i • d = d } | = | { d ∈ { + − } n ′ | τ i ′ • d = d } | = l = gcd ( i , n ) .Let n = l m , 2 ∤ m ∈ N , l ∈ N ∪ { } . Let i = l + j , gcd ( j , m ) = t >
1. Thenconsider n ′ = l mt , i ′ = l + jt . So i = ti ′ , n = tn ′ . We have by induction, using thetheorem for values smaller than n , | { d ∈ { + − } n ′ | ζ i ′ • d = d } | = gcd ( i ′ , n ′ ) = l .Let ( d , d , · · · , d t , d , d , · · · , d t , · · · , d n ′ , d n ′ , · · · , d tn ′ ) ∈ { + − } tn ′ . Weobserve that ζ ti ′ • ( d , d , · · · , d t , d , d , · · · , d t , · · · , d n ′ , d n ′ , · · · , d tn ′ )= ( d , d , · · · , d t , d , d , · · · , d t , · · · , d n ′ , d n ′ , · · · , d tn ′ ) if and only if τ i ′ • ( d j , d j , · · · , d jn ′ ) = ( d j , d j , · · · , d jn ′ ) for 1 ≤ j ≤ t .So we have | { d ∈ { + − } n | ζ i • d = d } | = | { d ∈ { + − } n ′ | τ i ′ • d = d } | t = l t = gcd ( i , n ) .Now let n = l m , 2 ∤ m ∈ N , l ∈ N ∪ { } . Let i = k j , 0 ≤ k ≤ l , 2 ∤ j . Herelet k =
0, that is, i is odd and suppose gcd ( i , n ) = ⇒ gcd ( i , 2 n ) =
1. If ( d , d , · · · , d n ) is a fixed point of ζ then we have ( d , d , · · · , d n ) = ( − d n , d , d , · · · , d n − ) ⇒ d = − d which is impossible. Hence there are no fixed pointsof ζ . Now let 1 ≤ i ′ ≤ n − ii ′ ≡ n . Then ζ i • d = d ⇒ ( ζ i ) i ′ • d = d ⇒ ζ • d = d which is impossible. Hence there are no fixed pointsof ζ i .Now let n = l m , 2 ∤ m ∈ N , l ∈ N ∪ { } . Let i = k j , 0 ≤ k ≤ l , 2 ∤ j . Here againlet k =
0, that is, i is odd and suppose 1 < gcd ( i , n ) = t ⇒ gcd ( i , 2 n ) = t . Thenwe consider n ′ = nt , i ′ = it and apply a similar argument as before, to concludethat there are no fixed points for ζ i as there are no fixed points for τ i ′ .Now let n = l m , 2 ∤ m ∈ N , l ∈ N ∪ { } . Let i = k j , 0 ≤ k ≤ l , 2 ∤ j . Here let k ≥
1. Then we consider n ′ = n , i ′ = i and apply a similar argument as beforeto conclude that there are fixed points for ζ i as there are no fixed points for τ i ′ .Hence Theorem . follows. (cid:4) C.P. ANIL KUMAR
Theorem . . The cardinality of the orbit space S n \S tnz n ( R ) is given by n ∑ k | n ,2 ∤ k φ ( k ) nk , for n ≥ where φ is the Euler-Totient function.Proof. First we observe that O M is conjugate to O tM = − O M = O N where N = MP − with P a permutation matrix in S n such that Pe ni = e nn − i + , 1 ≤ i ≤ n .So it is enough to count the number of orbits for the action of S n on the set { O M | M ∈ Mat tnz n ( R ) } of orientation sign matrices by conjugation.Now we observe that this action is isomorphic to the restricted action of P n to S n on the space of left cosets of K n ⊂ P n , that is, the map φ : { pK n | p ∈ P n } −→{ O M | M ∈ Mat tnz n ( R ) } = { pO n p − | p ∈ P n , O n is the standard orientationsign matrix } taking pK n to pO n p − is a well defined map and an isomorphismof the P n -sets. Also the set of combinatorial representations (a subset of P n ) ofelements in Mat tnz n ( R ) which give orientation sign matrix pO n p − is preciselythe right coset K n p − .The cardinality of the orbit space S n \S tnz n ( R ) is therefore given by1 n ! (cid:18) ∑ C a left coset of K n | Stab ( C ) | (cid:19) .If C = pK n then Stab ( C ) = S n ∩ pK n p − . Now let us find a system of distinctcoset representatives of K n in P n . A system of complete left coset representativesis precisely given by { ( a , a , · · · , a n ) | ( a , a , · · · , a n ) is a signed permutation of 2, 3, · · · , n } .This set also has the right cardinality 2 n − ( n − ) ! = | P n || K n | . Let D n be the subgroupof P n corresponding to the diagonal matrices in the group of signed permuta-tion matrices. Then we have D n ∼ = ( Z Z ) n = { + − } n and P n ∼ = S n ⋉ D n where S n is the subgroup of P n corresponding to permutation matrices. If p = ( a , a , · · · , a n ) ∈ P n the matrix associated to p is P where it is given by P = ( sgn ( a ) e n sgn ( a ) a , sgn ( a ) e n sgn ( a ) a , · · · , sgn ( a n ) e n sgn ( a n ) a n )= ( e n sgn ( a ) a , e n sgn ( a ) a , · · · , e n sgn ( a n ) a n ) . Diag ( sgn ( a ) , sgn ( a ) , · · · , sgn ( a n ))= QD where Q is a permutation matrix and D is a diagonal matrix.In the group P n this multiplication is expressed as p = ( a , a , · · · , a n )= ( sgn ( a ) a , sgn ( a ) a , · · · , sgn ( a n ) a n ) . ( sgn ( a )
1, sgn ( a ) · · · , sgn ( a n ) n )= qd where q ∈ S n , d ∈ D n . OINT ARRANGEMENTS AND STRATIFICATION OF TOTALLY NONZERO GRASSMANNIAN So p = qd ⇒| Stab ( pK n ) | = | S n ∩ pK n p − | = | p − S n p ∩ K n | = | d − S n d ∩ K n | = | dS n d ∩ K n | since d − = d . So we have1 n ! (cid:18) ∑ C a left coset of K n | Stab ( C ) | (cid:19) =( n − ) ! n ! (cid:18) ∑ ( d , d , ··· , d n ) ∈ ( Z Z ) n − | (
1, 2 d , 3 d , · · · , nd n ) S n (
1, 2 d , 3 d , · · · , nd n ) ∩ K n | (cid:19) = n (cid:18) ∑ d ∈ D n | dS n d ∩ K n | (cid:19) .Now let ψ : P n −→ S n be the surjective map with kernel D n given by ψ ( a , a , · · · , a n ) = ( sgn ( a ) a , sgn ( a ) a , · · · , sgn ( a n ) a n ) .Then ψ ( K n ) = h ( n , 1, 2, · · · , ( n − )) i is a cyclic subgroup order n generated bythe n -cycle ( · · · n ) in cycle notation. We have ψ ( q ) = q for q ∈ S n , ψ ( d ) = identity for d ∈ D n . For q ∈ S n , dqd ∈ K n ⇒ q = ψ ( q ) = ψ ( dqd ) = ( n , 1, · · · , ( n − )) i = ( n − i + n − i + · · · , n − n , 1, 2, · · · , n − i − n − i ) for some 0 ≤ i ≤ n − q = ( a , a , · · · , a n ) ∈ S n , d = ( d , 2 d , · · · , nd n ) ∈ D n where ( d , d , · · · , d n ) ∈ ( Z /2 Z ) n = { + − } n then we have dqd = ( a d d a , a d d a , · · · , a n d n d a n ) .So we have d ( n , 1, · · · , ( n − )) i d =(( n − i + ) d d n − i + , ( n − i + ) d d n − i + , · · · , ( n − ) d i − d n − , nd i d n ,1 d i + d , 2 d i + d , · · · , ( n − i − ) d n − d n − i − , ( n − i ) d n d n − i ) Hence for 0 ≤ i ≤ n − d ( n , 1, · · · , ( n − )) i d ∈ K n then either the followingholds:( . ) d ( n , 1, · · · , ( n − )) i d = ( − n , 1, 2, · · · , ( n − )) i and d j = − d n − i + j for 1 ≤ j ≤ i and d i + j = d j for 1 ≤ j ≤ n − i ,or the following holds:( . ) d ( n , 1, · · · , ( n − )) i d = ( − n , 1, 2, · · · , ( n − )) n + i and d j = d n − i + j for 1 ≤ j ≤ i and d i + j = − d j for 1 ≤ j ≤ n − i . C.P. ANIL KUMAR
Hence we consider the action of the cyclic group Z n Z = < ζ > on the group ( Z /2 Z ) n = { + − } n as follows. The action of the generator ζ is given by ζ • ( d , d , · · · , d n ) = ( − d n , d , d , · · · , d n − ) .In terms of this group action Equations . imply ζ i • ( d , d , · · · , d n ) = ( d , d , · · · , d n ) .In terms of this group action Equations . imply ζ n + i • ( d , d , · · · , d n ) = ( d , d , · · · , d n ) .So we have1 n ! (cid:18) ∑ C a left coset of K n | Stab ( C ) | (cid:19) = n (cid:18) ∑ d ∈ D n | dS n d ∩ K n | (cid:19) = n (cid:18) ∑ d ∈{ + − } n | { i | ζ i • d = d , 0 ≤ i ≤ n − } | (cid:19) = n (cid:18) n − ∑ i = | { d ∈ { + − } n | ζ i • d = d } | (cid:19) .We prove the theorem using Theorem . . Now for an integer j we have 1 ≤ j ≤ m − ⇐⇒ ≤ l + j < l + m = n . Let k | m , k =
1. The cardinality of the set { j | ≤ j ≤ m − gcd ( j , m ) = mk } is exactly φ ( k ) where φ is the Euler-totientfunction. Hence the cardinality of the set { i | ≤ i ≤ n − gcd ( i , n ) = nk , i = l + j } is φ ( k ) . So we have1 n ! (cid:18) ∑ C a left coset of K n | Stab ( C ) | (cid:19) = n (cid:18) n − ∑ i = | { d ∈ { + − } n | ζ i • d = d } | (cid:19) = n (cid:18) n + ∑ k | m , k = φ ( k ) nk (cid:19) = n ∑ k | n ,2 ∤ n φ ( k ) nk .This completes the proof of the theorem. (cid:4) Remark . . The initial values for 2 ≤ n ≤
10 are given as 1, 2, 2, 4, 6, 10, 16, 30, 52.Also refer OEIS Sloane sequence A [ ]. OINT ARRANGEMENTS AND STRATIFICATION OF TOTALLY NONZERO GRASSMANNIAN . Open Questions: Enumeration of the Generic Point Arrangements, theAntipodal Point Arrangements in Higher Dimensions
Theorem . gives parameter spaces for the isomorphism classes of the genericpoint arrangements of cardinality n and the isomorphism classes of the antipo-dal point arrangements of cardinality 2 n in the Euclidean space R m . In Sec-tions . , . , we have obtained combinatorial representations of a generic pointarrangement of cardinality n in the plane and of an element in Mat tnz n ( R ) whichlead to the enumeration of parameter spaces for m =
2. We also have observedthat there is a single isomorphism class for the antipodal point arrangements ofcardinality n in the plane in Theorem . .The analogous enumeration questions about the cardinalities of these parame-ter spaces (( R ∗ ) n ⋊ S n ) \S tnzmn ( R ) and S n \S tnzmn ( R ) are still open for m ≥
3. Theenumeration of the space S tnzmn ( R ) of strata for the totally nonzero Grassmannianis also open for m ≥
3. It is known that there is more than one isomorphismclass for the antipodal point arrangements in the space R of cardinality 2 n for n = ], [ ]) even though it is not a difficult exerciseto show that there is a single isomorphism class of the antipodal point arrange-ments in the space R of cardinality 2 n for each n =
3, 4, 5. The antipodal pointarrangements in general are combinatorially classified in [ ].R eferences [ ] C. P. Anil Kumar, On infinity type hyperplane arrangements and convex positive bijections , Nov. , pages, https://arxiv.org/pdf/1711.07030.pdf [ ] C. P. Anil Kumar, Antipodal point arrangements on spheres and classification of normal systems ,Nov. , pages, https://arxiv.org/pdf/1801.09575.pdf [ ] N. E. Mn¨ev, The universality theorems of the classification problem of configuration varieties andconvex polytope varieties , Topology and geometry-Rohlin Seminar, pp. - , Lecture Notesin Math., , Springer, Berlin, Heidelberg, ., https://doi.org/10.1007/BFb0082792 ,MR [ ] OEIS Foundation Inc., Sequence A , The On-Line Encyclopedia of Integer Sequences, https://oeis.org/A000016 [ ] A. Postnikov, Total positivity, Grassmannians and networks , Preprint, , arXiv:math/ [ ] L. K. Williams, Enumeration of totally positive Grassmann cells , Advances in Mathematics,Vol. , Issue , Jan. , pp. - , https://doi.org/10.1016/j.aim.2004.01.003 ,MR P ost D octoral F ellow in M athematics , H arish -C handra R esearch I nstitute , C hhatnag R oad , J hunsi , P rayagraj (A llahabad )- , U ttar P radesh , INDIA Email address ::