aa r X i v : . [ m a t h . A C ] A ug DIAGONAL DEGENERATIONS OF MATRIX SCHUBERT VARIETIES
PATRICIA KLEIN
Abstract.
Knutson and Miller (2005) established a connection between the anti-diagonalGr¨obner degenerations of matrix Schubert varieties and the pre-existing combinatorics ofpipe dreams. They used this correspondence to give a geometrically-natural explanationfor the appearance of the combinatorially defined Schubert polynomials as representativesof Schubert classes. Recently, Hamaker, Pechenik, and Weigandt proposed a similar con-nection between diagonal degenerations of matrix Schubert varieties and bumpless pipedreams, newer combinatorial objects introduced by Lam, Lee, and Shimozono. Hamaker,Pechenik, and Weigandt described new generating sets of the defining ideals of matrixSchubert varieties and conjectured a characterization of permutations for which these gen-erating sets are diagonal Gr¨obner bases. They proved special cases of this conjecture anddescribed diagonal degenerations of matrix Schubert varieties in terms of bumpless pipedreams in these cases. The purpose of this paper is to prove the general conjecture. Theproof uses a connection between liaison and geometric vertex decomposition establishedin earlier work with Rajchgot. Introduction
Schubert polynomials, introduced by Lascoux and Sch¨utzenberger [23] based on the workof Bernstein, Gel’fand, and Gel’fand [3], give combinatorially-natural representatives ofSchubert classes in the cohomology ring of the complete flag variety. Matrix Schubert vari-eties, defined by Fulton [11], are generalized determinantal (affine) varieties correspondingto a permutation w ∈ S n . A key insight of Knutson and Miller [18] is that the combinatoricsof Schubert polynomials is naturally reflected in the geometry of the initial ideals of thedefining ideals I w of matrix Schubert varieties X w under anti-diagonal degeneration (i.e.,Gr¨obner degeneration under a term order in which the lead term of the determinant of ageneric matrix is the product of the entries along the anti-diagonal). In particular, Knut-son and Miller were able to identify irreducible components of anti-diagonal initial schemeswith the pipe dreams that had arisen in earlier combinatorial study of Schubert polynomials[2, 10]. They were also able to give a geometric explanation for the positivity of coefficientsof Schubert polynomials, a fact not obvious from their recursive definition, and show thatthe multidegrees of X w give the torus-equivariant cohomology classes of Schubert varieties.Later, Knutson, Miller, and Yong [19] connected the geometry of diagonal degenerations(defined analogously to anti-diagonal degenerations) of matrix Schubert varieties corre-sponding to vexillary permutations to the combinatorics of flagged tableaux. And morerecently, Hamaker, Pechenik, and Weigandt proposed in [16] a diagonal Gr¨obner basis,which they call the set of CDG generators (see Subsection 2.2), for a wider class of permu-tations that includes the vexillary permutations. They proved that CDG generators forma diagonal Gr¨obner basis when w is banner and used that result to connect the geometryof the diagonal degenerations of X w to the bumpless pipe dreams introduced by Lam, Lee,and Shimozono [21] (closely related to the 6-vertex ice model used by Lascoux [22, 24, 5]).With an eye towards extending their main theorem [16, Theorem 6.4], Hamaker, Pechenik,and Weigandt made the following conjecture: onjecture 1.1. [16, Conjecture 7.1] Let w ∈ S n be a permutation. The CDG generatorsare a diagonal Gr¨obner basis for I w if and only if w avoids all eight of the patterns , , , , , , , . The purpose of the present paper is to prove Conjecture 1.1, which we do as Corollaries3.17 and 4.2. An important step in our proof is an application of the author’s work withRajchgot [17, Corollary 4.13], which uses the connection between liaison , whose use instudying Gr¨obner bases is described in [15], and geometric vertex decomposition , introducedin [19], to essentially reduce the requirements of the liaison-theoretic approach to a checkon one ideal containment.The appearance of pattern avoidance to determine when CDG generators form a Gr¨obnerbasis is natural in light of similar results in the Schubert literature. For example, patternavoidance has previously been seen to govern the singularity [20] and Gorenstein property[25] of Schubert varieties as well as when the
Fulton generators (see Subsection 2.1) of I w constitute a diagonal Gr¨obner basis [19]. (See also [14] for one direction of this last result inthe language of mixed ladder determinantal varieties.) For a survey of results in this vein,see [1].In [16], the authors note that Conjecture 1.1 implies the following conjecture by the workof [9]: Conjecture 1.2. [16, Conjecture 7.2]
If the (single) Schubert polynomial of w ∈ S n is amultiplicity-free sum of monomials, then the CDG generators of I w are a diagonal Gr¨obnerbasis. We refer the reader to [16, 24] for a more information on Schubert polynomials and bumplesspipe dreams.
The structure of this paper:
Section 2 is devoted to preliminaries on matrix Schubertvarieties and CDG generators. In Section 3, we prove the backward direction of Conjec-ture 1.1, and, in Section 4, we prove the forward direction. Finally, in Section 5, we usegeometric vertex decomposition to give some intuition on what unifies the eight non-CDGpermutations listed in Conjecture 1.1.
Acknowledgements:
The author thanks Zach Hamaker, Oliver Pechenik, and AnnaWeigandt for helpful conversations and for graciously sharing their L A TEXcode for Rothediagrams. She is also grateful to Jenna Rajchgot for many very valuable conversationsboth directly concerning this paper and also on related material. She thanks all four forcomments on an earlier draft of this document.2.
Preliminaries
In this section, we review the basics of matrix Schubert varieties as well as the CDGgenerators introduced in [16]. For a more detailed introduction to matrix Schubert varieties,we refer the reader to [12, Chapter 10]. For basic properties of and standard terminologyon Gr¨obner bases, we refer the reader to [8].2.1.
Matrix Schubert varieties.
We begin by describing how each permutation is asso-ciated to the affine variety called a matrix Schubert variety. Throughout this paper, wewill take [ n ] = { , , . . . , n } for any n ≥ S n denote the symmetric group on [ n ].Each permutation w ∈ S n is is a bijection w : [ n ] → [ n ], which we will record in its one-line otation w = w w . . . w n where w i = w ( i ). To every w ∈ S n , we associate a Rothe diagram D w , defined as follows: D w = { ( i, j ) ∈ [ n ] × [ n ] : w ( i ) > j, w − ( j ) > i } . A Rothe diagram has the following visualization: In an n × n grid, place a • in posi-tion ( i, w i ) for each i ∈ [ n ], and draw a line down from each • to the bottom of thegrid and a line to the right from each • to the side of the grid. Then D w is set ofboxes in the grid without a • in them or a line through them. For example, D is { (1 , , (1 , , (3 , , (3 , , (4 , , (4 , , (5 , } and corresponds to the visualization below, inwhich the elements of D appear in gray and will be referred to as the boxes of w : . The
Coxeter length of the permutation w is equal to its inversion number, i.e. |{ ( i, j ) | i < j, w i > w j }| , which is in turn equal to | D w | . For example, the Coxeter length of 315642 is 7, easily readoff as the number of gray boxes in the diagram above. Definition 2.1.
Fix a permutation w = w · · · w n ∈ S n and a permutation v = v . . . v k ∈ S k with k ≤ n . If there is some substring w i · · · w i k of w satisfying w i j < w i ℓ exactly when v j < v ℓ , we say that w contains v . Otherwise, we say that w avoids v . (cid:3) For example, w = 13254 contains v = 2143 with 3254 the substring of w realizing thecontainment, but w does not contain v ′ = 3214. Notice that if w i · · · w i k satisfies w i j < w i ℓ exactly when v j < v ℓ , then the Rothe diagram of v can be obtained from that of w byrestricting to the rows i , . . . , i k and columns w i , . . . , w i k in the [ n ] × [ n ] grid giving thevisualization of D w .By restricting to the maximally southeast boxes of D w , we define the essential set of w :Ess( w ) = { ( i, j ) ∈ D w | ( i + 1 , j ) , ( i, j + 1) / ∈ D w } . In the example above, Ess(315642) = { (1 , , (4 , , (5 , } . Borrowing a term from theliterature on ladder determinantal varieties, if ( i, j ) ∈ Ess( w ) and there is no ( i ′ j ′ ) ∈ Ess( w )strictly southeast of ( i, j ), we will say that ( i, j ) is a lower outside corner of D w . In thecase of w = 315642, (4 ,
4) and (5 ,
2) are lower outside corners, but (1 ,
2) is not.To every permutation w ∈ S n , we associate a rank function rank w : [ n ] × [ n ] → Z , whererank w ( i, j ) = |{ k ≤ i | w ( k ) ≤ j }| and the rank matrix M w whose ( i, j ) th entry is rank w ( i, j ). Visually, we assign to everysquare ( i, j ) in the [ n ] × [ n ] grid underlying the Rothe diagram of w the number of • s weaklynorthwest of ( i, j ). In our running example w = 315642, one may find it helpful to record he information as M w = or as 0 0111 22 .We have recorded elements of D w in the rank matrix M w as boxes and colored the boxesof essential set orange in anticipation of our discussion of Fulton generators, below.Let X = ( x i,j ) ( i,j ) ∈ [ n ] × [ n ] be a matrix of distinct indeterminates and R = C [ X ] so thatSpec( R ) is the affine n -space Mat n,n of complex n × n matrices. For subsets A, B ⊆ [ n ],let X A,B = ( x i,j ) i ∈ A,j ∈ B be the submatrix of X determined by the rows whose index is anelement of A and the columns whose index is an element of B . Then the matrix Schubertvariety of w ∈ S n is the affine variety X w = (cid:8) X ∈ Mat n,n | rank X [ i ] , [ j ] ≤ rank w ( i, j ) for all ( i, j ) ∈ [ n ] × [ n ] (cid:9) , which is defined by the Schubert determinantal ideal I w = ((rank w ( i, j ) + 1)-minors in X [ i ] , [ j ] | ( i, j ) ∈ [ n ] × [ n ]) ⊆ R. This naive generating set will typically include a good deal of redundancy, and so we willmore often consider the smaller set of
Fulton generators of I w : { (rank w ( i, j ) + 1)-minors in X [ i ] , [ j ] | ( i, j ) ∈ Ess( w ) } . Fulton showed that I w is prime and, in particular, that X w ∼ = Spec( R/I w ) as reducedschemes [11, Proposition 3.3]. The height of the ideal I w (equivalently, codimension ofSpec( R/I w ) in Spec( R )) is equal to the Coxeter length of w .In the example w = 315642, the Fulton generators of I w are x , , x , , the 2-minors of x , x , x , x , x , x , x , x , x , x , , and the 3-minors of x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , . CDG generators.
In [16], the authors introduce
CDG generators of defining ideals ofmatrix Schubert varieties. These generators are named after Conca, De Negri, and Gorla,whose result [7, Theorem 4.2] served as inspiration for the generating set used in [16] and,in particular, for Conjecture 1.1.
Definition 2.2.
Fix a permutation w ∈ S n and an n × n matrix X = ( x i,j ) ( i,j ) ∈ [ n ] × [ n ] ofdistinct indeterminates. Let Dom( w ) = { ( i, j ) ∈ D w | rank w ( i, j ) = 0 } , and call Dom( w )the dominant part of the Rothe diagram D w . From X , form the matrix X ′ by replacing x i,j by 0 whenever ( i, j ) ∈ Dom( w ). Set G ′ w = { (rank w ( i, j ) + 1)-minors in X ′ [ i ] , [ j ] | ( i, j ) ∈ Ess( w ) \ Dom( w ) } , and G w = G ′ w ∪ { x i,j | ( i, j ) ∈ Dom( w ) } . We call G w the set of CDG generators of I w . (cid:3) Example 2.3. If w = 315642 the CDG generators of I w are x , , x , , the 2-minors of x , x , x , x , x , x , x , x , , and the 3-minors of x , x , x , x , x , x , x , x , x , x , x , x , x , x , . (cid:3) Notice that Dom( w ) = ∅ if and only if w = 1, in which case the CDG generators and theFulton generators coincide. When G w forms a Gr¨obner basis for I w under every diagonalterm order, we will say that w is CDG .3.
Rothe diagrams of CDG permutations
Obstructions to being CDG.
We begin this section by describing in terms of theRothe diagram D w conditions that prevent w from being CDG. In Subsection 3.2, we willshow that when D w does not satisfy these conditions, w is necessarily CDG.Before we begin, we note that the visualization of the Rothe diagram of 214635 is ob-tained from that of 215364 by transposition. The same is true of 315264 and 241635. Thevisualizations of the Rothe diagrams of the remaining permutations listed in Conjecture1.1 are self transpose. This symmetry will allow us to consolidate some of our case workbelow. Throughout the remainder of this paper, we will understand the cardinal directionsin reference to D w in terms of its visualization. We will say, for example, that ( i ′ , j ′ ) is“strictly southeast” of ( i, j ) to mean that both i ′ > i and also j ′ > j , or that ( i ′ , j ′ ) is“strictly south and weakly east” of ( i, j ) to mean i ′ > i and also j ′ ≥ j . Definition 3.1.
The permutation w has an obstruction of • Type 1 if there is some ( r, s ) ∈ Dom( w ) ∩ Ess( w ) and two distinct entries ( i, j ) and( i ′ , j ′ ) of D w strictly southeast of ( r, s ) with i ′ = i and j ′ = j , • Type 2 if there is some ( r, s ) ∈ Dom( w ) ∩ Ess( w ) and two distinct entries ( i, j ) and( i, j ′ ) of Ess( w ) strictly southeast of ( r, s ) withmax k { ( k, j ) ∈ Dom( w ) } = max k { ( k, j ′ ) ∈ Dom( w ) } or, symmetrically, two distinct entries ( i, j ) and ( i ′ , j ) of Ess( w ) strictly southeastof ( r, s ) with max ℓ { ( i, ℓ ) ∈ Dom( w ) } = max ℓ { ( i ′ , ℓ ) ∈ Dom( w ) } , • Type 3 if there are two distinct entries ( i, j ) and ( i ′ , j ) of Ess( w ) \ Dom( w ) with( i ′ , j ′ ) strictly southeast of ( i, j ). (cid:3) Lemma 3.2.
If the permutation w ∈ S n has an obstruction of Type , then w contains , , , , or .Proof. Fix a permutation w ∈ S n that has an obstruction of Type 1, and fix entries ( r, s ),( i, j ), and ( i ′ , j ′ ) as in the definition of an obstruction of Type 1. Consider the visualizationof the Rothe diagram D w .Label the • in the column s + 1 with ( a, w a ) and the • in row r + 1 with ( b, w b ). Notice a < b and w a > w b . Because ( i, j ) and ( i ′ , j ′ ) are strictly southest of ( r, s ), both x i,j and x i ′ ,j ′ must be south of row b and east of column w a . We consider two orientations of ( i, j )and ( i ′ , j ′ ). irst, if ( i, j ) is strictly northeast of ( i ′ , j ′ ), then we label the • in row i with ( c, w c ) andthe • in column j ′ with ( d, w d ). Now a < b < c < d and w b < w a < w d < w c . (Here c = i and w d = j ′ . Similar renamings will occur below.) If there is any • in any row strictlybetween c and d whose column index is strictly between w d and w c , choose one and nameit ( e, w e ). Then we will have a < b < c < e < d and w b < w a < w d < w e < w c , which is tosay that w contains 21543. Otherwise, the • in row i ′ , which we call ( f, w f ), must be eastof column w c , and the • in column j , which we call ( g, w g ), must be south of row d , and so a < b < c < f < d < g and w b < w a < w d < w f < w c < w e , which is to say that w contains215634.Alternatively, if ( i, j ) is strictly northwest of ( i ′ , j ′ ), we label the • in row i ′ with ( c, w c ),the • in column j ′ with ( d, w d ), the • in row i with ( e, w e ), and the • in column j with( f, w f ). If w e > w c , then a < b < e < c < d while w b < w a < w d < w c < w e , so w contains21543. Similarly, if f < d , then w also contains 21543. Hence, we may now assume thateither w e < w d < w c or w d < w e < w c and either f < c < d or c < f < d . Each of these fourpossibilities require the containment of 215634, 215364, 214635, or 13254 (ignoring ( b, w b )in case w e < w d < w c and f < c < d and by using all six dots in the other three cases). (cid:3) Example 3.3.
We give an illustration of the case ( i, j ) strictly northeast of ( i ′ , j ′ ) todemonstrate the process of considering allowable regions of the visualization of D w for • swe know must exist but whose location is unknown. Either at least one the • s in column j and row i ′ fall in Region I (in blue), or both fall in Region II (in green). If the former,then w contains 21543, and, if the latter, then w contains 215634.Dom( w )( r, s ) ( i, j )( i ′ , j ′ ) I IIII . (cid:3)
Lemma 3.4.
If the the permutation w ∈ S n contains an obstruction of Type , then w contains , , , , or .Proof. Fix a permutation w ∈ S n that has an obstruction of Type 2. We will first assumethat we have fixed ( i, j ) and ( i, j ′ ) as in the definition of a Type 2 obstruction with j ′ < j and ( r, s ) assumed to be the easternmost element of Dom( w ) ∩ Ess( w ) northwest of both( i, j ) and ( i, j ′ ). As before, we consider the visualization of the Rothe diagram D w .As before, label the • in column s + 1 with ( a, w a ) and the • in the row r + 1 with( b, w b ). Notice a < b and w a > w b . Label the • in row i with ( c, w c ), the • in column j with ( d, w d ), and the • in column j ′ with ( e, w e ). If e > d , then we have a < b < c < d < e and w b < w a < w e < w d < w c , which is to say that w contains 21543. Now assume e < d . Because ( i ′ , j ′ ) , ( i, j ) ∈ Ess( w ), there must be some • in column j ′ + 1, which welabel ( f, w f ), north of row i . Because of the easternmost assumption on ( r, s ) and because ax k { ( k, j ) ∈ Dom( w ) } = max k { ( k, j ′ ) ∈ Dom( w ) } , we must have that w f > w a . If row f > b > a , then w contains 214635 and, if b > f > a , then w contains 241635.A parallel argument show that if i ′ < i , j ′ = j , and max ℓ { ( i, ℓ ) ∈ Dom( w ) } = max ℓ { ( i ′ , ℓ ) ∈ Dom( w ) } , then w contains 21543, 215364, or 315264. (cid:3) Lemma 3.5.
If the permutation w ∈ S n has an obstruction of Type , then w contains , , , , , , , or .Proof. Fix a permutation w ∈ S n that has an obstruction of Type 3. We fix ( i, j ), ( i ′ , j ) asin the definition of an obstruction of Type 3, and consider the visualization of the Rothediagram of w . If there is some ( r, s ) ∈ Dom( w ) ∩ Ess( w ) strictly northwest of ( i, j ), then w has an obstruction of Type 1, and so it follows from Lemma 3.2 that w contains 21543,215634, 215364, 214635, or 13254. Hence, we assume no such ( r, s ) exists. Without loss ofgenerality, we assume that the • in row 1 is west of column j ′ and that the • in column 1is north of row i ′ .First suppose that the • in row 1 is west of column j . Then the assumption that thereis no ( r, s ) ∈ Dom( w ) ∩ Ess( w ) strictly northwest of ( i, j ) implies that the • in column 1 issouth of row i . Because ( i, j ) ∈ Ess( w ), there must be a • in column j + 1 weakly north ofrow i . If the • in column j is north of row i ′ , then the • s in row 1 and columns j and j + 1combine with the • s in row i ′ and column j ′ to form 13254. If the • in column j is south ofthe • in column j ′ , then they combine with the • s in row 1, column 1, and row i ′ to form21543. And if the • in column j is north of the • in column j ′ but south of row i ′ , then all • s described in this paragraph form 241635.Alternatively, assume that the • in row 1 is between columns j and j ′ . If that the • incolumn 1 is north of row i , then transposing the argument in the previous paragraph showsthat w must contain 13254, 21543, or 315264. If the • in column 1 is south of row i , labelthe • in row 1 with ( a, w a ), any fixed • northwest of ( i, j ) with ( b, w b ), and the • in column1 with ( c, w c ). We know that there is some • northwest of ( i, j ) because ( i, j ) / ∈ Dom( w ).As before, if the • in row i is west of column j ′ and the • in column j is north of row i ,then w contains 13254. Suppose that the • in row i is east of column j ′ , and label that • with ( d, w d ). Label the • in row i ′ with ( e, w e ), and the • in column j ′ with ( f, w f ). If w e < w d , then ( a, w a ), ( b, w b ), ( d, w d ), ( e, w e ), and ( f, w f ) form 21543. If w e > w d , then weconsider the placement of the • in column j , which we label ( g, w g ). If g < e , then ( a, w a ),( b, w b ), ( d, w d ), ( e, w e ), ( f, w f ), and ( g, w g ) form 315264. If e > g > f , then all • s ( a, w a ) to( g, w g ), form 4261735. And if f < g , then ( b, w b ), ( c, w c ), ( e, w e ), ( f, w f ), and ( g, w g ) form21543. Finally, the cases in which w d < w f (equivalently, the • in row i west of column j ′ )and g < e follow by symmetry. (cid:3) Permutations avoiding the specified patterns are CDG.
The remainder of thissection is devoted to the backward direction of Conjecture 1.1. The framework will be tobuild to a use of [17, Corollary 4.13]. We begin with some notation.If I w is the Schubert determinantal ideal of the permutation w ∈ S n , we will use X w to denote the matrix obtained from an n × n matrix of indeterminates by setting x i,j to 0whenever ( i, j ) ∈ Dom( w ). If ( i, j ) is a lower outside corner of D w and y = x i,j , we writethe CDG generators of I w as { yq + r , . . . , yq k + r k , h , . . . , h ℓ } where y does not divideany term of any q i , r i or h j . Define N y,I w = ( h , . . . , h ℓ ) and C y,I w = ( q , . . . , q k , h , . . . , h ℓ ).This notation mimics that in [17]. When the CDG generators are a Gr¨obner basis of I w , C y,I w will be the ideal corresponding to the star and N y,I w + ( y ) the ideal corresponding tothe link in a geometric vertex decomposition in the sense of [19].We will call { q , . . . , q k , h , . . . , h ℓ } the CDG generators of C y,I w , which is itself not typ-ically a Schubert determinantal ideal. With notation as above, we begin by showing that y,I w is the Schubert determinantal ideal of a permutation whose Coxeter length is smallerthan that of w , which will be an essential component of an inductive argument. Lemma 3.6.
Suppose that I w is the Schubert determinantal ideal of the permutation w ∈ S n and that ( i, j ) is a lower outside corner of D w corresponding to the variable y = x i,j . Theideal N y,I w is the Schubert determinantal ideal of a permutation w ′ ∈ S n whose Coxeterlength is strictly smaller than that of w satisfying D w ′ ( D w .Proof. We claim that whenever there is some (rank w ( i, j ) + 1)-minor with yq + r = r ,that r ∈ N y,I w , i.e., that all of the CDG generators of I w determined only by the rankcondition at ( i, j ) involve y . Fix a (rank w ( i, j ) + 1) × (rank w ( i, j ) + 1) submatrix Z of X w so that det( Z ) = yq + r , and suppose that q = 0. Then q is the determinant of arank w ( i, j ) × rank w ( i, j ) submatrix of Z with an element of Dom( w ) along its anti-diagonal.If r = 0, then there is some ( i ′ , j ) with nonvanishing x i ′ ,j · q ′ summand of r so that q ′ is thedeterminant of a rank w ( i, j ) × rank w ( i, j ) submatrix of Z without an element of Dom( w )along its anti-diagonal. Because Dom( w ) forms a partition shape, if x i ′ ,j · q ′ = 0, then x i ′′ ,j · q ′′ = 0 whenever i ′′ < i ′ and q ′′ is the cofactor corresponding to x i ′′ ,j in an expansionof yq + r along column j . In particular, there is a unique t so that no rank w ( i, j ) × rank w ( i, j )submatrix of Z that excludes column j and involves the final t rows has a 0 along its anti-diagonal and every rank w ( i, j ) × rank w ( i, j ) submatrix of Z that excludes column j and alsoexcludes one of the final t rows has a 0 along its anti-diagonal. The same argument can beapplied to columns, and, because vanishing is determined by 0’s along the anti-diagonal,will select the final (rank w ( i.j )+1 − t ) columns. Hence, we may write r as the product of one t -minor determined by the final t rows and initial t columns of Z and one (rank w ( i.j )+1 − t )-minor consisting of the initial rank w ( i.j ) + 1 − t rows and final rank w ( i.j ) + 1 − t columnsof Z . Call the southeast corner of the lower block z and the southeast corner of the upperblock z ′ . If there are fewer than t dots northwest of z , then the t -minor that is one factor of r is an element of N y,I w , and so r ∈ N y,I w . If there are t or more dots northwest of z , thenthere are at most rank w ( i.j ) − t dots northwest of z ′ , and so the factor of r correspondingto the upper block is an element of N y,I w and so r ∈ N y,I w . Hence, N y,I w is generated bythe CDG generators of I w determined by the essential boxes other than ( i, j ). In particular,if w ′ is obtained from w by setting w ′ ℓ = w ℓ when ℓ = i, w − ( j ), w ′ i = j , and w ′ w − ( j ) = w i ,then N y,I w = I w ′ , w ′ has smaller Coxeter length than w , and D w = D w ′ ∪ ( i, j ) (with( i, j ) D w ′ ). (cid:3) Remark 3.7.
With notation as in Lemma 3.6, while it is possible that Ess( w ′ ) Ess( w ),as is the case if w = 215634 and ( i, j ) = (4 , w ′ = 215436, and (3 , , (4 , ∈ Ess( w ′ ) \ Ess( w ). In general, it is easy to see from the construction of w ′ that the onlypossible elements of Ess( w ′ ) \ Ess( w ) are ( i − , j ) and ( i, j − (cid:3) Corollary 3.8. If w has no obstruction of Type , , or , ( i, j ) is a lower outside cornerof D w corresponding to the variable y = x i,j , and I w ′ = N y,I w , then w ′ has no obstructionof Type , , or .Proof. The claim concerning Type 1 obstructions is immediate from the fact that D w ′ ( D w ,and the claims concerning Types 2 and 3 follow quickly from the restrictions on Ess( w ′ ) \ Ess( w ) in Remark 3.7. (cid:3) Next, we will show that the CDG generators of C y,I w form a Gr¨obner basis whenever w has no obstruction of Type 1, 2, or 3. Before proceeding, we review some standard notationand make one new definition to help with bookkeeping during this subsection. With respectto a fixed term order, we will denote the lead term, or initial term, of a polynomial f by T ( f ) and let LT ( I ) = ( LT ( f ) | f ∈ I ) denote the initial ideal of I . We will use deg( f ) todenote the degree of the homogeneous polynomial f and LCM ( µ , µ ) to denote the leastcommon multiple of two monomials (which will arise for us as the monic lead terms of idealgenerators). Definition 3.9.
Fix a permutation w ∈ S n , lower outside corner ( i, j ) of D w correspondingto the variable y = x i,j in X w , and CDG generators { yq + r , . . . , yq k + r k , h , . . . , h ℓ } of I w ,where y does not divide any q a , r a , or h b . Assume also that there is some 0 ≤ ℓ ′ ≤ ℓ so that allvariables appearing in h b are northwest of some ( i ′ , j ) ∈ Ess( w ) with rank w ( i ′ , j ) = deg( h b )or of some ( i, j ′ ) with rank w ( i, j ′ ) = deg( h b ) if and only if b ≤ ℓ ′ . If (1 , j ) ∈ Dom( w ),set m = min { i − p | ( p, j ) ∈ Dom( w ) } and set m = i if (1 , j ) / ∈ Dom( w ). Similarly, if( i, ∈ Dom( w ), set m = min { j − q | ( i, q ) ∈ Dom( w ) } and set m = j if (1 , j ) / ∈ Dom( w ).We form the ideal Q y,I w = ( ( q , . . . , q k ) rank w ( i, j ) + 1 = min { m , m } ( q , . . . , q k , h , . . . , h ℓ ′ ) otherwise (cid:3) Less formally, we taking Q y,I w = ( q , . . . , q k ) when the rank condition on ( i, j ) is deter-mining maximal minors in the submatrix of X w obtained from the submatrix northwest of x i,j by deleting any full rows or columns of 0’s and, otherwise, including also as generatorsof Q y,I w the CDG generators determined by essential boxes in the same row or columnas y . We make this definition purely for technical convenience below and not out of anindependent interest in Spec( R/Q y,I w ). We will say that Q y,I w is CDG if the generatorsgiven above form a Gr¨obner basis under any diagonal term order. Lemma 3.10. If w ∈ S n has no obstruction of Type , , or and Conjecture 1.1 holds forall permutations of smaller Coxeter length than that of w , then there is some lower outsidecorner ( i, j ) of D w corresponding to the variable y in X w so that Q y,I w is CDG.Proof. Fix a permutation w ∈ S n that has no obstruction of Type 1, 2, or 3. First supposethat D w has some lower outside corner ( i, j ) corresponding to the variable y in X w satisfyingrank w ( i, j )+1 = min { m , m } , with notation as in Definition 3.9. Write the CDG generatorsof I as { yq + r , . . . , yq k + r k , h , . . . , h ℓ } , where y does not divide any q i , r i , or h j . Asdiscussed in Lemma 3.6, every rank w ( i, j ) + 1-minor involving row i and column j has aterm divisible by y , and so { q , . . . , q k } generates the ideal of rank w ( i, j )-minors in thesubmatrix of X w strictly northwest of y . Because the Q y,I is an ideal of maximal minors(after possible removing full rows or columns of 0’s), the result follows from [7, Theorem4.2] or [4, Proposition 5.4].Alternatively, suppose that D w has no such lower outside corner, and fix any lower outsidecorner of D w . Because Q y,I w depends only on D w weakly northwest of ( i, j ), we may assumethat ( i, j ) is the only lower outside corner of D w and that (1 , j ) , ( i, / ∈ Dom( w ). Withthis assumption, Q y,I w = C y,I w , and it will suffice to show that I w is CDG, from whichit follows from [19, Theorem 2.1(a)] that C y,I w is CDG. If Dom( w ) = ∅ , then, because w has no obstruction of Type 1, w must be vexillary, and so the desired result is that of [19,Theorem 3.8]. Otherwise, the assumptions that (1 , j ) , ( i, / ∈ Dom( w ) imply that there isat least one element of Dom( w ) ∩ Ess( w ) northwest of ( i, j ). Choose the southernmost suchelement and label it ( r, s ) and the easternmost element and label it ( r ′ , s ′ ). Because w hasno obstruction of Type 1, all essential boxes of D w must be in either row i or column j .Because rank w ( i, j ) + 1 < min { i, j } , there must be at least one essential box in row i andat least one in column j aside from ( i, j ). Then because w has no obstruction of Type 2, here are no elements of Ess( w ) north of row i and south of row r or west of column j andeast of column s ′ .We will argue directly in this case that for each pair q a and h b , their s -polynomial hasa Gr¨obner reduction by the CDG generators of I w . Choose such a q a and h b . Becausethe CDG generators of N y,I w form a Gr¨obner basis by induction on the Coxeter lengthof w and Corollary 3.8, we may assume that q a / ∈ ( h , . . . , h ℓ ). Because I w involves onlyindeterminates of X w northwest of x i,j , we will work within the submatrix of X w northwestof x i,j , which we will call Y w . Suppose that h b is determined by the essential box at ( i ′ , j ) forsome i ′ < i . (The case of ( i, j ′ ) with j ′ < j will follow by symmetry.) We consider two cases.First, suppose that ( i ′ , j − ∈ D w . Then, because w has no obstruction of Type 1, therecan be no element of Dom( w ) ∩ Ess( w ) northwest of ( i ′ , j ). In particular, r ′ ≥ i ′ , and h b isa (rank w ( i ′ , j ) + 1)-minor in the submatrix of Y w formed of its final j − s ′ columns, which isa generic matrix and which we will call Y ′ w . Suppose that there are t columns determining q a strictly east of column s ′ and rank w ( i, j ) + 1 − t columns determining q a weakly west ofcolumn s ′ . Express q a as a sum of products of t -minors and (rank w ( i, j ) + 1 − t )-minorscorresponding to this subdivision. For any fixed set of rows of Y ′ w , the set of (rank w ( i ′ , j )+1)-minors northwest of ( i ′ , j ) together with the t -minors northwest of ( i, j ) in the submatrixof Y ′ w including only those specified rows forms a Gr¨obner basis for the ideal they generatebecause it is a mixed ladder determinantal ideal [14, Theorem 1.10]. Choose the t -minor ε from the western t columns determining q a and the rank w ( i, j ) + 1 − t -minor ε fromthe remaining columns satisfying LT ( ε ) · LT ( ε ) = LT ( q a ). Because ε , h b belong to theideal of rank w ( i, j ) + 1 − t -minors weakly north of its southernmost entries together with therank w ( i ′ , j ) + 1-minors northwest of ( i ′ , j ) in Y ′ w , their s -polynomial s ( ε , h b ) has a Gr¨obnerreduction s ( ε , h b ) = P α c δ c by the natural generators of that ideal. For each δ c ∈ N y,I w ,set b δ c = LT ( ε ) · δ c , and for each δ c involving some row south of i ′ , set b δ c to be (up to sign)the determinant of the augmentation of the matrix determining δ c by the rows and columnsdetermining ε (with sign chosen so that LT ( δ c ) and LT ( b δ c ) share a sign). Let s ( q a , h b )denote the s -polynomial of q a and h b .We claim that s ( q a , h b ) − P α c b δ c ∈ N y,I . It is clear that s ( q a , h b ) − P α c b δ c contains a LT ( ε )-multiple of s ( ε , h b ) − P α c δ c , which is 0 because s ( ε , h b ) − P α c δ c is. Fix anynon-leading term µ of ε , and write s ( q a , h b ) − P α c b δ c = µs ′ + s ′′ where µ does not divideany term of s ′′ . For each column involved in ε , whenever a different variable from thatcolumn divides µ and LT ( ε ), replace in P α c b δ c the variables in the row of the divisor of LT ( ε ) with the variables in the same columns from the row of the corresponding divisorof µ , which we note gives an expression s ′ in terms of the natural generators of the ideal of(rank w ( i ′ , j ) + 1)-minors northwest of ( i ′ , j ), each of which is a CDG generator of N y,I .Now because LT ( α c δ c ) ≤ LT ( s ( ε , h b )) and LT ( α c b δ c ) = LT ( ε ) · LT ( α c δ c ) for each c andbecause LT ( s ( q a , h b )) = LT ( ε ) · LT ( s ( ε , h b )), subtracting each α c b δ c is a valid step in aGr¨obner reduction of s ( q a , h b ) by the generators of I w . The fact that the CDG generatorsof N y,I w , each of which is a CDG generator of I w , form a Gr¨obner basis for the ideal theygenerate implies that s ( q a , h b ) − P α c b δ c ∈ N y,I w has a reduction in terms of those generatorsand so that s ( q a , h b ) has a reduction by the CDG generators of I w .In the second case, in which ( i ′ , j − / ∈ D w , there can be no j ′ < j with ( i ′ , j ′ ) ∈ D w because w has no obstruction of Type 3. Hence, rank w ( i ′ , j ) + 1 = min { j − k | ( i ′ , k ) ∈ Dom( w ) } , and so the (rank w ( i ′ , j ) + 1)-minors northwest of ( i ′ , j ) are the maximal minorsof the submatrix of X w northwest of ( i ′ , j ) after removing complete rows or columns of0’s. Then the argument is similar to the first case but uses, instead of results on ladder eterminantal ideals in a generic matrix, the fact that the maximal minors of matrices ofindeterminates and 0’s form a Gr¨obner basis by [7, Theorem 4.2] or [4, Proposition 5.4.]. (cid:3) Example 3.11.
Observe that w = 36718245, whose annotated Rothe diagram appearsbelow, is an example of a CDG permutation with a unique lower outside corner ( i, j )and rank w ( i, j ) + 1 = min { i, j } . We illustrate how the reduction of the s -polynomial of h b = (cid:12)(cid:12)(cid:12)(cid:12) x , x , x , x , (cid:12)(cid:12)(cid:12)(cid:12) and (cid:12)(cid:12)(cid:12)(cid:12) x , x , x , x , (cid:12)(cid:12)(cid:12)(cid:12) gives rise to that of (cid:12)(cid:12)(cid:12)(cid:12) x , x , x , x , (cid:12)(cid:12)(cid:12)(cid:12) and q a = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x , x , x , x , x , x , x , x , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) by the process described in Lemma 3.10. Using that x , (cid:12)(cid:12)(cid:12)(cid:12) x , x , x , x , (cid:12)(cid:12)(cid:12)(cid:12) − x , (cid:12)(cid:12)(cid:12)(cid:12) x , x , x , x , (cid:12)(cid:12)(cid:12)(cid:12) + x , (cid:12)(cid:12)(cid:12)(cid:12) x , x , x , x , (cid:12)(cid:12)(cid:12)(cid:12) + x , (cid:12)(cid:12)(cid:12)(cid:12) x , x , x , x , (cid:12)(cid:12)(cid:12)(cid:12) = 0 , we construct the equation x , x , (cid:12)(cid:12)(cid:12)(cid:12) x , x , x , x , (cid:12)(cid:12)(cid:12)(cid:12) + x , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x , x , x , x , x , x , x , x , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − x , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x , x , x , x , x , x , x , x , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + x , x , (cid:12)(cid:12)(cid:12)(cid:12) x , x , x , x , (cid:12)(cid:12)(cid:12)(cid:12) = x , (cid:18) x , (cid:12)(cid:12)(cid:12)(cid:12) x , x , x , x , (cid:12)(cid:12)(cid:12)(cid:12) + x , (cid:12)(cid:12)(cid:12)(cid:12) x , x , x , x , (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) . In the latter equation, we find an x , -multiple of the first equation (whose terms appearin blue) and an x , -multiple of an element easily seen to be in N y,I w . We record in orangethe generators of N y,I w that give the inclusion of the x , -multiple summand of s ( q a , h b ) in N y,I w and see that relation arising from the first equation. The new relation is obtainedfrom the first by exchanging rows 4 and 5.0 00 00 01 2 21 11 1 . (cid:3) Before proceeding, we recall one very useful lemma.
Lemma 3.12. ( [6, Lemma 1.3.14] ) Let I and J be homogeneous ideals of a polynomial ringover a field, and fix a term order σ . With respect to σ , let F be a Gr¨obner basis of I and G a Gr¨obner basis of J . Then F ∪ G is a Gr¨obner basis of I + J if and only if for all f ∈ F and g ∈ G there exists e ∈ I ∩ J such that LT ( e ) = LCM ( LT ( f ) , LT ( g )) . (cid:3) We are now prepared to show that, under a suitable inductive hypothesis, there is a loweroutside corner so that the CDG generators of C y,I w are Gr¨obner. Lemma 3.13.
With notation as above, if w ∈ S n avoids obstructions of Types , , and and Conjecture 1.1 holds for all permutations of smaller Coxeter length than that of w ,then there is some lower outside corner ( i, j ) of D w corresponding to the variable y = x i,j so hat the generators { q , . . . , q k , h , . . . , h ℓ } of C y,I w form a Gr¨obner basis under any diagonalterm order.Proof. Fix a diagonal term order σ . By Lemma 3.10, there is some lower outside cornerof D w so that, with notation as above, the generators { ( q , . . . , q k , h , . . . , h ℓ ′ ) } (for some ℓ ′ ≥
0) form a Gr¨obner basis for Q y,I w . By Corollary 3.8 and the inductive hypothesis, { h , . . . , h ℓ } is a diagonal Gr¨obner basis for N y,I w . Then by Lemma 3.12, the generators { q , . . . , q k , h , . . . , h ℓ } form a diagonal Gr¨obner basis for C y,I w if and only if for every q a and h b there exists some f ∈ ( q , . . . , q k , h , . . . , h ℓ ′ ) ∩ ( h , . . . , h ℓ ) satisfying LT ( f ) = LCM ( LT ( q i ) , LT ( h j )).If h b ∈ ( q , . . . , q k ) or q a ∈ ( h , . . . , h ℓ ), the result follows from Lemma 3.12, and if LCM ( LT ( q a ) , LT ( h b )) = LT ( q a ) · LT ( h b ), then we take f = q a · h b . Otherwise, there issome ( r, s ) ∈ Ess( w ) \ Dom( m ) with rank w ( r, s ) = deg( h b ) − e = x r,s weakly southeast of all of the variables involved in h b . Because w has no Type 3obstruction, e is not strictly northwest of y . Suppose first that y is strictly east and weaklynorth of e .In this case, let M ′ be the matrix consisting of the union of the columns determining q a and h b and the union of the rows determining q a and h b . Form a matrix M from M ′ asfollows: First set to 0 any entry whose row index is not one of the rows determining q a andwhose column index is not one of the columns determining h b . Next, whenever a columnof M ′ contains a leading term from q a and a distinct leading term from h b , duplicate thatcolumn and replace in one copy of the column the variables coming only from h b by 0.Whenever a row of M ′ contains a leading term from q a and a distinct leading term from h b ,duplicate that row and replace in one copy of the row the variables coming only from q a by0. Now M will be a d × d matrix where d = deg( q a ) + deg( h b ) − deg( LCM ( LT ( q a ) , LT ( h b )))because it will have one row and one column for each monomial dividing LT ( q a ) and oneeach for every monomial dividing LT ( h b ) but not LT ( q a ).By expressing det( M ) as a sum of products of the deg( q a )-minors from the rows of M originating from the submatrix of X w determining q a and the ( d − q a )-minors in theremaining rows, we see that det( M ) ∈ ( q , . . . , q k ). Similarly, by expressing det( M ) as asum of products of deg( h b )-minors in the column originating from the submatrix of X w determining from h b and ( d − deg( h b ))-minors in the remaining columns, we have det( M ) ∈ ( h , . . . , h ℓ ). It is because y is strictly east and weakly north of e that the rows determining q a that every deg( q a )-minor in the specified rows is an element of ( q , . . . , q k ) and that everydeg( h b )-minor in the specified column is an element of ( h , . . . , h ℓ ).Now we must see that LT (det( M )) = LCM ( LT ( q a ) , LT ( h b )). Call f M the submatrix of M ′ whose entries are northwest of both e and y . Set µ to be the product of the termsof f M that divide either LT ( q a ) or LT ( h b ). Call µ the lead term of the determinant ofthe submatrix of M consisting of the rows and columns used to determine h b excludingthose involving a divisor of µ . Similarly, define µ to be the lead term of the determinantof the submatrix of M consisting of the rows and columns determining q a excluding thoseinvolving a divisor of µ . Notice that µ · µ · µ = LCM ( LT ( q a ) , LT ( h b )) and that thisproduct is a term of det( M ). To see that every other term of det( M ) is smaller under σ ,notice that because w has no obstruction of Type 1, the submatrix of M ′ whose entries areboth northwest of e and northwest of y must have 0’s only in full rows and full columns alongthe north and west sides. It will now be more convenient for us to work with c M , obtainedfrom M by adding the doubled copies of rows and columns obtained in the transition from M ′ to M so that no variable appears more than once. Note that det( c M ) = det( M ). Ifsome other term of det( c M ) is larger than µ · µ · µ , there must be some entries of c M ividing µ · µ · µ whose row indices we may permute to obtain a larger monomial. Wemay assume that this permutation consists of one cycle. If all entries divide either µ · µ or µ · µ , we would obtain a term of q a or h b , respectively, that is strictly larger thanits leading term, which also cannot be. But the permutation cannot send any divisor of µ to the row of a divisor of µ , all of which are 0 in that column, or vice versa. Hence, µ · µ · µ = LT (det( c M )) = LT (det( M )), as desired.Finally, if y is strictly south and weakly west of e , a parallel argument gives the result. (cid:3) Example 3.14.
Below we give an example of the construction of the matrices M ′ and M if w = 5237164, y = x , , q a = det x , x , x , x , x , x , x , ∈ C y,I w , and h b = det x , x , x , x , x , x , x , x , x , x , x , x , x , x , ∈ N y,I w , in which case LT ( q a ) = x , x , x , , LT ( h b ) = x , x , x , x , , and LCM ( LT ( q a ) , LT ( h b )) = x , x , x , x , x , x , = LT (det( M )) . The variables dividing the leading terms appearing throughout this example are noted inblue. Then M ′ = x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , and M = x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , By expressing det( M ) as a sum of products of 3-minors in the first 3 rows of M with3-minors in the final 3 rows, we see det( M ) ∈ C y,I w , and. by expressing det( M ) as the sumof products of 4-minors in columns 1, 2, 4, and 5 with 2-minors in columns 3 and 6, we seedet( M ) ∈ N y,I w . Observe that LT (det( M )) = LT (cid:18) det (cid:20) x , x , x , x , (cid:21)(cid:19) · LT det x , x , x , x , x , x , x , x , · x , · = ( x , x , )( x , x , x , )( x , ) . This expression corresponds to the product µ · µ · µ in the theorem above. One mayprefer to use c M , as in the theorem, obtained from M by subtracting column 3 from column4, which has the effect of setting the copies of x , and x , in column 4 to 0.One aspect of this example that is typical of the general case is the intersection of thesubmatrix of of M ′ northwest of x , (playing the role of e ) and that northwest of x , (from x , playing the role of y ) is a rectangular matrix of indeterminates with 0’s appearing onlyin full rows along the top full columns along the western side of M ′ . This arrangementfollows from the fact that 5237164 has no obstruction of Type 1 and gives rise to thedecomposition of det( M ) described above into a product of the lead term of entries alongthe main diagonal of a matrix of indeterminates together with entries coming only from q a and entries coming only from h b . (cid:3) We are now prepared to use our lemmas above that establish Gr¨obner bases for N y,I w and C y,I w by induction to give the backward direction of Conjecture 1.1. We will recall a emma that structures our proof below. This lemma requires the notion of a y -compatibleterm order, as described though unnamed in [19]. If y is a variable in the polynomial ring R , we say that a term order σ is y -compatible if, for every f ∈ R written f = y k q + r where y does not divide any term of q and y k does not divide any term of r , LT ( f ) = LT ( y k q ). Lemma 3.15. [17, Corollary 4.13] . Let I = ( yq + r , . . . , yq k + r k , h , . . . , h ℓ ) be a ho-mogenous ideal of the polynomial ring R with y some variable of R and y not dividing anyterm of any q i nor any term of any h j . Fix a y -compatible term order σ , and suppose that G C = { q , . . . , q k , h , . . . , h ℓ } and G N = { h , . . . , h ℓ } are Gr¨obner bases for the ideals theygenerate, which we call C and N , respectively, and that ht ( I ) , ht ( C ) > ht ( N ) . Assume that N has no embedded primes. Let M = (cid:18) q · · · q k r · · · r k (cid:19) . If the ideal of -minors of M iscontained in N , then the given generators of I are a Gr¨obner basis. Theorem 3.16. If w ∈ S n is a permutation that has no obstruction of Type , Type , orType , then w is CDG.Proof. Fix a diagonal term order σ . We proceed by induction on the Coxeter length of w ,with the case of length 0 trivial. We fix a permutation w ∈ S n with n arbitrary and assumethat w has no obstruction of Type 1, Type 2, or Type 3According to Lemma 3.13, there is some lower outside corner ( i, j ) of D w corresponding tothe variable y = x i,j so that, with our usual notation, the generators { q , . . . , q k , h , . . . , h ℓ } of C y,I w form a Gr¨obner basis. We claim first that it is sufficient to consider y -compatibleterm orders. Define a new term order σ ′ as follows: For any two monomials µ and µ , µ < σ ′ µ if max t { y t | µ } < max t { y t | µ } or if max t { y t | µ } = max t { y t | µ } and µ < σ µ . Because ( i, j ) is a lower outside corner, y divides exactly once the σ -lead termof every CDG generator of I w in which it is involved. Hence, an s -polynomial reduction oftwo CDG generators computed with respect to σ ′ remains valid under σ . In particular, ifthe CDG generators form a Gr¨obner basis under σ ′ , they form a Gr¨obner basis under σ ,which is to say that we may assume that σ is y -compatible.By Corollary 3.8 and the inductive hypothesis, { h , . . . , h ℓ } is a Gr¨obner basis for N y,I w .We will show that I (cid:18) q . . . q k r . . . r k (cid:19) ⊆ N y,I . For each CDG generator, yq a + r a , let yq ′ a + r ′ a be the corresponding natural generator of I w , i.e. the generator taken in a matrix ofindeterminates in which the variables corresponding to Dom( w ) have not been set to 0.Let J = ( x i,j | ( i, j ) ∈ Dom( w )). Then I (cid:18) q . . . q k r . . . r k (cid:19) + J = I (cid:18) q ′ . . . q ′ k r ′ . . . r ′ k (cid:19) + J .Hence, because J ⊆ N y,I w , it suffices to show I (cid:18) q ′ . . . q ′ k r ′ . . . r ′ k (cid:19) ⊆ N y,I w . Now because each q ′ a r ′ b − q ′ b r ′ a = ( yq ′ a + r ′ a ) q ′ b − ( yq ′ b + r ′ b ) q ′ a for 1 ≤ a < b ≤ k is an element of the ideal of(rank w ( i, j ) + 1)-minors in a matrix of indeterminates weakly northwest of y , an ideal forwhich the natural generators form a diagonal Gr¨obner basis, we know that q ′ a r ′ b − q ′ b r ′ a has aGr¨obner reduction in terms of those generators. Because q ′ a r ′ b − q ′ b r ′ a does not involve y and σ is assumed to be y -compatible, that reduction must be in terms of (rank w ( i, j )+ 1)-minorsweakly northwest of y that do not involve y . Each such generator is an element of N y,I w .Hence, I (cid:18) q . . . q k r . . . r k (cid:19) + J = I (cid:18) q ′ . . . q ′ k r ′ . . . r ′ k (cid:19) + J ⊆ N y,I w , as desired. he height requirements ht I w , ht C y,I w > N y,I w are immediate from the fact that N y,I w is prime [11, Proposition 3.3] together with the proper containment of N y,I w in each of I w and C y,I w . The result now follows from Lemma 3.15. (cid:3) Notice that we do not claim that a Gr¨obner reduction of q ′ a r ′ b − q ′ b r ′ a in terms of thenatural generators gives rise to a Gr¨obner reduction of a a r b − q b r a in terms of the CDGgenerators. Lemma 3.15 requires only that we demonstrate an ideal containment. Corollary 3.17. If w ∈ S n avoids all eight of the following patterns, then w is CDG: , , , , , , , . Proof. If w ∈ S n avoids the patterns above, then it does not have an obstruction of Type1, Type 2, or Type 3 by Lemmas 3.2, 3.4, and 3.5, and so the result follows from Theorem3.16. (cid:3) The non-CDG Permutations
In this section, we first show that a permutation w ∈ S n that contains one of the eightpermutations listed in Conjecture 1.1 is not CDG. Theorem 4.1. If w ∈ S k is not CDG and v ∈ S n +1 contains w for some k ≤ n , then v isnot CDG.Proof. By induction, we may assume that w = ( w . . . w n ) ∈ S n and that v = ( v . . . v n +1 )with v . . . v i − v i +1 . . . v n +1 = w for some 1 ≤ i ≤ n + 1. Fix some diagonal term order σ with respect to which the CDG generators of I w are not Gr¨obner. Recall that D w isobtained from D v by deleting row i and column v i . With X v an n + 1 × n + 1 matrix ofindeterminates with x i,j set to 0 whenever ( i, j ) ∈ Dom( v ), identify X w with the n × n submatrix of X v obtained by the deletion of row i and column v i . Notice that the rows of X w are labeled 1 , . . . , i − , i + 1 , . . . n + 1 and the columns 1 , . . . , v i − , v i +1 , . . . , n + 1. Let G w = { δ , . . . , δ ℓ } for some ℓ ∈ N be the set of CDG generators of w , and assume that G w isordered so that δ . . . . , δ k are determined by rank conditions in boxes ( a, b ) ∈ Ess( w ) with a < i or b < v i and that δ k +1 , . . . , δ ℓ are determined by rank conditions in boxes ( a, b ) with a > i and b > v i . Let f and g be two CDG generators of I w whose s -polynomial s = s ( f, g )does not reduce to 0 under σ . Let r denote the remainder of s under the deterministicdivision algorithm with respect to G w and the chosen ordering on G w . Then we may write r = s + P α j δ j where the leading term of α j δ j is not in the ideal generated by the leadingterms of the δ j ′ with j ′ < j . By definition of remainder, no leading term of any element of G w divides the leading term of r though r ∈ I w .Let G v denote the set of CDG generators of I v . We may write G v = { δ , . . . , δ k , x i,v i δ k +1 + ε k +1 , . . . , x i,v i δ ℓ + ε ℓ , δ ℓ +1 , . . . , δ m } , where the δ j with ℓ < j ≤ m are the elements of G v involving at least one variable from row i or column v i other than x i,v i , and the others areas expected. We will use r to construct an element of I v whose leading term is not divisibleby any leading term of G v . If the southeast corner of the submatrix of X w determining f is a box ( a, b ) satisfying a < i or b < v i , then f ∈ G v . In that case, define f ′ = f . If a > i and b > v i , then take f ′ to be the element of G v determined by the rows determining f together with row i and the columns determining f together with column v i . In particular, f ′ = x i,v i f + ε f , where every term of ε f is divisible by exactly one variable from row i andexactly one variable from column v i , neither of which is x i,v i . Define g ′ similarly, and take s ′ = s ( f ′ , g ′ ) to be their s -polynomial. If f ′ = f and g ′ = g , then s ′ = s . Because no termof s is divisible by any variable in row i or in column v i of X v , if s has a reduction by theelements of G v , it must have a reduction by { δ , . . . , δ k } , which is known not to exist. f f ′ = x i,v i f + ε f and g ′ = g , let LT ( f ) denote the leading term of f , LT ( g ) denote theleading term of g , and G the greatest common divisor of LT ( f ) and LT ( g ). Set t = LT ( g ) G f ′ − x i,v i LT ( f ) G g = x i,v i s + LT ( g ) G ε f ∈ I v . (Notice that whenever x i,v i LT ( f ) is the leading term of f ′ , t will coincide with the s -polynomial of f ′ and g ′ .) We claim that t cannot be reduced by G v . We begin by modifying t by multiples of the δ i for 1 ≤ i ≤ k following the deterministic division algorithm in G w to obtain t ′ = x i,v i r + LT ( g ) G ε f ∈ I v . Notice that LT ( g ) G ε f does not involve x i,v i and thatevery element of G v involving x i,v i involves it only as a multiple of some δ j with k < i ≤ ℓ .Hence, no term of any δ j divides any term of x i,v i r , and the division algorithm will nevercall for the addition of any multiple of any x i,v i δ j . Therefore, no newly added polynomialcould have any term that cancels with any term of x i,v i r , from which it follows that t ′ is anelement of I v with no reduction by G v .Finally, assume that f ′ = x i,v i f + ε f and g ′ = x i,v i g + ε g . Then t = LT ( g ) G f ′ − LT ( f ) G g ′ = s + LT ( g ) G ε f − LT ( f ) G ε g ∈ I v . Again, we modify by multiples of the δ j with 1 ≤ j ≤ k to obtain t ′ = r + LT ( g ) G ε f − LT ( f ) G ε g .Because no leading term of any δ j with 1 ≤ j ≤ k divides any term of r and because everyterm of every other element of G v involves a variable from for i or column v i , which r doesnot, no further steps in the division algorithm can eliminate any term of r , and so t has noreduction by the elements of G v .It follows that in all cases, there is an element of I v that has no Gr¨obner reduction by G v , and so G v is not a diagonal Gr¨obner basis of I v . (cid:3) Corollary 4.2.
Let w be a permutation. If the CDG generators are a diagonal Gr¨obnerbasis for I w , then w avoids all eight of the patterns , , , , , , , . Proof.
This result is immediate from Theorem 4.1 together with explicit computations inthe case of the eight permutations listed in Conjecture 1.1. (cid:3) Unifying characteristics of the non-CDG permutations
We conclude by describing briefly how [19, Theorem 2.1(a)] can be used to understand 2properties that prevent the permutations listed in Conjecture 1.1 from being CDG. We notefirst that 13254 has no dominant part and so its failure to be CDG is due to the fact thatit contains 2143 [19, Theorem 6.1]. For the remainder of this section, we consider the other7 permutations, all of which have nontrivial dominant parts. For an arbitrary rank matrix,understand the CDG generators to be defined analogously to the case of defining ideals ofmatrix Schubert varieties. If any of the permutations listed in Conjecture 1.1 were CDG,[19, Theorem 2.1(a)] would require that either the ideal determined by the rank matrix N or the ideal determined by the rank matrix N , below, have a CDG Gr¨obner basis, whichthey are easily seen not to: N = and N = . he rank matrix N encodes interference from Dom( w ) that prevents I w from beingCDG, and N encodes failures to be vexillary that are sufficiently far from Dom( w ) thatthey are not handled by replacing Fulton generators by CDG generators. Example 5.1.
Consider the rank matrix M w of the permutation w = 21543 with respectto any y = x , -compatible term order, with essential boxes marked by (cid:3) . M w = If the CDG generators of I w were a Gr¨obner basis, then [19, Theorem 2.1(a)] wouldrequire the CDG generators of C y,I w , which is the ideal determined by N and plays therole of the link in a geometric vertex decomposition at y , also to be a Gr¨obner basis. (cid:3) We leave it to the reader to use (possibly repeated) application of [19, Theorem 2.1(a)]to obtain N or N from the rank matrices of the other 6 permutations listed in Conjecture1.1. References [1] H. Abe and S. Billey,
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University of Minnesota, School of Mathematics, Minneapolis, MN, USA.
E-mail address , Patricia Klein: [email protected]@umn.edu