Basic quadratic identities on quantum minors
aa r X i v : . [ m a t h . QA ] F e b Basic quadratic identities on quantum minors
Vladimir I. Danilov ∗ Alexander V. Karzanov † Abstract
This paper continues an earlier research of the authors on universal quadratic iden-tities (QIs) on minors of quantum matrices. We demonstrate situations when theuniversal QIs are provided, in a sense, by the ones of four special types (Pl˝ucker,co-Pl˝ucker, Dodgson identities and quasi-commutation relations on flag and co-flaginterval minors).
Keywords : quantum matrix, Pl˝ucker and Dodgson relations, quasi-commuting mi-nors, Cauchon graph, path matrix
MSC-class : 16T99, 05C75, 05E99
Let A be a K -algebra over a field K and let q ∈ K ∗ . We deal with an m × n matrix X whoseentries x ij belong to A and satisfy the following “quasi-commutation” relations (originallyappeared in Manin’s work [9]): for i < ℓ ≤ m and j < k ≤ n , x ij x ik = qx ik x ij , x ij x ℓj = qx ℓj x ij , (1.1) x ik x ℓj = x ℓj x ik and x ij x ℓk − x ℓk x ij = ( q − q − ) x ik x ℓj . We call such an X a fine q -matrix over A and are interested in relations in the cor-responding quantized coordinate ring (the algebra of polynomials in the x ij respecting therelations in A ), which are viewed as quadratic identities on q - minors of X . Let us start withsome terminology and notation. • For a positive integer n ′ , the set { , , . . . , n ′ } is denoted by [ n ′ ]. Let E n,m denote theset of ordered pairs ( I, J ) such that I ⊆ [ m ], J ⊆ [ n ] and | I | = | J | ; we will refer to such apair as a cortege and may denote it as ( I | J ). The submatrix of X whose rows and columnsare indexed by elements of I and J , respectively, is denoted by X ( I | J ). For ( I, J ) ∈ E m,n ,the q - determinant (called the q - minor , the quantum minor ) of X ( I | J ) is defined as∆ X,q ( I | J ) := X σ ∈ S k ( − q ) ℓ ( σ ) Y kd =1 x i d j σ ( d ) , (1.2) ∗ Central Institute of Economics and Mathematics of RAS, 47, Nakhimovskii Prospect, 117418 Moscow,Russia; email: [email protected] † Central Institute of Economics and Mathematics of RAS, 47, Nakhimovskii Prospect, 117418 Moscow,Russia; email: [email protected]. Corresponding author. Q are ordered from left to right by increasing d , and ℓ ( σ ) denotes the length (number of inversions) of a permutation σ . The terms X and/or q in ∆ X,q ( I | J ) maybe omitted when they are clear from the context. By definition ∆( ∅|∅ ) is the unit of A . • A quantum quadratic identity (QI) of our interest is viewed as X (sign i q δ i ∆ q ( I i | J i ) ∆ q ( I ′ i | J ′ i ) : i = 1 , . . . , N ) = 0 , (1.3)where for each i , δ i ∈ Z , sign i ∈ { + , −} , and ( I i | J i ) , ( I ′ i | J ′ i ) ∈ E m,n . Note that anypair ( I | J ) , ( I ′ | J ′ ) may be repeated in (1.3) many times. We restrict ourselves by merely homogeneous QIs, which means that in expression (1.3),(1.4) each of the sets I i ∪ I ′ i , I i ∩ I ′ i , J i ∪ J ′ i , J i ∩ J ′ i is invariant of i .When, in addition, (1.3) is valid for all appropriate A , q, X (with m, n fixed), we say that (1.3)is universal .In fact, there are plenty of universal QIs. For example, representative classes involvingquantum flag minors were demonstrated by Lakshmibai and Reshetikhin [6] and Taft andTowber [11]. Extending earlier results, the authors obtained in [4] necessary and sufficientconditions characterizing all universal QIs. These conditions are given in combinatorial termsand admit an efficient verification.Four special cases of universal QIs play a central role in this paper. They are exposed in(I)–(IV) below; for details, see [4, Sects. 6,8].In what follows, for integers 1 ≤ a ≤ b ≤ n ′ , we call the set { a, a + 1 , . . . , b } an interval in[ n ′ ] and denote it as [ a..b ] (in particular, [1 ..n ′ ] = [ n ′ ]). For disjoint subsets A and { a, . . . , b } ,we may abbreviate A ∪ { a, . . . , b } as Aa . . . b . Also for ( I | J ) ∈ E m,n , ∆( I | J ) = ∆ X,q ( I | J ) iscalled a flag ( co-flag ) q -minor if J = [ k ] (resp. I = [ k ]), where k := | I | = | J | .(I) Pl˝ucker-type relations with triples . Let A ⊂ [ m ], B ⊂ [ n ], { i, j, k } ⊆ [ m ] − A , ℓ ∈ [ n ] − B , and let | A | + 1 = | B | and i < j < k . There are several universal QIs on suchelements (see a discussion in [4, Sect. 6.4]). One of them is viewed as∆( Aj | B )∆( Aik | Bℓ ) = ∆( Aij | Bℓ )∆( Ak | B ) + ∆( Ajk | Bℓ )∆( Ai | B ) . (1.5)In the flag case (when B = [ | B | ] and ℓ = | B | + 1) this turns into a quantum analog of theclassical Pl˝ucker relation with a triple i < j < k .(II) Co-Pl˝ucker-type relations with triples . They are “symmetric” to those in (I). Namely,we deal with A ⊂ [ m ], B ⊂ [ n ], ℓ ∈ [ m ] − A and { i, j, k } ⊆ [ n ] − B such that | A | = | B | + 1and i < j < k . Then there holds:∆( A | Bj )∆( Aℓ | Bik ) = ∆( Aℓ | Bij )∆( A | Bk ) + ∆( Aℓ | Bjk )∆( A | Bi ) . (1.6)(III) Dodgson-type relations . Let i, k ∈ [ m ] and j, ℓ ∈ [ n ] satisfy k − i = ℓ − j ≥
0. Formthe intervals A := [ i + 1 ..k −
1] and B := [ j + 1 ..ℓ − Ai | Bj )∆( Ak | Bℓ ) = ∆( Aik | Bjℓ )∆( A | B ) + q ∆( Ai | Bℓ )∆( Ak | Bj ) . (1.7)2n particular, when A = B = ∅ , we obtain the expression ∆( ik | jℓ ) = ∆( i | j )∆( k | ℓ ) − q ∆( i | ℓ )∆( k | j ) (with k = i + 1 and ℓ = j + 1), taking into account that ∆( ∅|∅ ) = 1. Thismatches formula (1.2) for the q -minor of a 2 × Quasi-commutation relations on interval q -minors . The simplest possible kind ofuniversal QIs involves two corteges ( I | J ) , ( I ′ | J ′ ) ∈ E m,n and is viewed as∆( I | J )∆( I ′ | J ′ ) = q c ∆( I ′ | J ′ )∆( I | J ) (1.8)for some c ∈ Z . When q -minors ∆( I | J ) and ∆( I ′ | J ′ ) satisfy (1.8), they are called quasi-commuting . (For example, three relations in (1.1) are such.) Leclerc and Zelevinsky [7]characterized such minors in the flag case, by showing that ∆( I | [ | I | ]) and ∆( I ′ | [ | I ′ | ]) quasi-commute if and only if the subsets I, I ′ of [ m ] are weakly separated (for a definition, see [7]).In a general case, a characterization of quasi-commuting q -minors is given in Scott [10] (seealso [4, Sect. 8.3] for additional aspects).For purposes of this paper, it suffices to consider only interval q -minors , i.e., assume thatall I, J, I ′ , J ′ are intervals. Let for definiteness | I | ≥ | I ′ | and define α := |{ i ′ ∈ I ′ : i ′ < min( I ) }| , β := |{ i ′ ∈ I ′ : i ′ > max( I ) }| , (1.9) γ := |{ j ′ ∈ J ′ : j ′ < min( J ) }| , δ := |{ j ′ ∈ J ′ : j ′ > max( J ) }| . Then the facts that
I, J, I ′ , J ′ are intervals and that | I | ≥ | I ′ | imply αβ = γδ = 0.Specializing Proposition 8.2 from [4] to our case, we obtain that(1.10) for | I | ≥ | I ′ | , interval q -minors ∆( I | J ) and ∆( I ′ | J ′ ) quasi-commute (universally) if andonly if αγ = βδ = 0; in this case, c as in (1.8) is equal to β + δ − α − γ .In fact, we will use (1.10) only when ∆( I | J ) is a flag or co-flag interval q -minor, andsimilarly for ∆( I ′ | J ′ ) (including mixed cases with one flag and one co-flag q -minors).In this paper we explore the issue when the special quadratic identities exhibited in (I)–(IV) determine all other universal QIs. More precisely, let P = P m,n , P ∗ = P ∗ m,n , and D = D m,n denote the sets of relations as in (1.5), (1.6), and (1.7), respectively (concerningthe corresponding objects in (I)–(III)). Also let Q = Q m,n denote the set of quasi-commutingrelations in (IV) concerning the flag and co-flag interval cases. Definitions.
For A , q, m, n as above, f : E m,n → A is called a QI-function if its valuessatisfy the quadratic relations similar to those in the universal QIs on q -minors (i.e., whenwe formally replace ∆( I | J ) by f ( I | J ) in these relations). When f : E m,n → A is assumed tosatisfy the relations as in P , P ∗ and D , we say that f is an RQI-function (abbreviating “afunction obeying restricted quadratic identities ”).Note that if f : E m,n → A satisfies a quadratic relation Q, and a is an element of thecenter of A (i.e. ax = xa for any x ∈ A ), then af satisfies Q as well. Hence if f is a QI- orRQI-function, then so is af . Due to this, in what follows we will default assume that anyfunction f on E m,n we deal with is normalized , i.e., satisfies f ( ∅|∅ ) = 1 (which is consistentwith ∆( ∅|∅ ) = 1).Our goal is to prove two results on QI-functions. Let us say that a cortege ( I | J ) ∈ E m,n is a double interval if both I, J are intervals. A double interval ( I | J ) is called pressed if atleast one of I, J is an initial interval, i.e., either I = [ | I | ] or J = [ | J | ] or both (yielding a flagor co-flag case); the set of these is denoted as P int = P int m,n .3 heorem 1.1 Let RQI-functions f, g : E m,n → A − { } coincide on P int m,n . Let, inaddition, for any ( I | J ) ∈ E m,n , the element f ( I | J ) is not a zerodivisor in A . Then f and g coincide on the entire E m,n . It follows that any QI-function is uniquely determined by its values on
P int and relationsas in P , P ∗ and D .The second theorem describes a situation when taking values on P int arbitrarily withina representative part of A , one can extend these values to a QI-function (so one may saythat, P int plays a role of “basis” for QI-functions, in a sense).
Theorem 1.2
Let f : P int → A ∗ (where A ∗ is the set of invertible elements of A ). Supposethat f satisfies the quasi-commutation relations (as in (1.8) in (IV)) on P int . Then f isextendable to a QI-function f on E m,n . It should be noted that Theorems 1.1 and 1.2 can be regarded as quantum analogs ofcorresponding results in [5] devoted to universal quadratic identities on minors of matricesover a commutative semiring (e.g. over R > or over the tropical semiring ( R , + , max)); seeTheorem 7.1 there.This paper is organized as follows. Section 2 contains a proof of Theorem 1.1. Section 3reviews a construction, due to Casteels [2], used in our approach to proving the secondtheorem. According to this construction (of which idea goes back to Cauchon diagramsin [3]), the minors of a generic q -matrix can be expressed as the ones of the so-called pathmatrix of a special planar graph G m,n , viewed as an extended square grid of size m × n .There is a one-to-one correspondence between the pressed interval corteges in E m,n and theinner vertices of G m,n . This enables us to assign each generator involved in the constructionof entries of the path matrix (formed in Lindstr˝om’s style via path systems, or “flows”, in G m,n ) as the ratio of two values of f ; this is just where we use that f takes values in A ∗ .Relying on this construction, we prove Theorem 1.2 in Section 4; here the crucial step isto show that the quasi-commutation relations on the values of f imply the relations ongenerators needed to obtain a corrected path matrix. Finally, in Section 5 we describe asituation when a function f on P int m,n exposed in Theorem 1.2 has a unique extension to E m,n that is a QI-function, or, roughly speaking, when the values on P int and relations asin P , P , D and Q determine a QI-function on E m,n , thus yielding all other universal QIs. Let f, g : E m,n → A be as in the hypotheses of this theorem. To show that f ( I | J ) = g ( I | J )holds everywhere, we consider three possible cases for ( I | J ) ∈ E m,n . In the first and secondcases, we use induction on the value σ ( I, J ) := max( I ) − min( I ) + max( J ) − min( J ) . Case 1 . Let ( I | J ) be such that: (i) f ( I ′ | J ′ ) = g ( I ′ | J ′ ) holds for all ( I ′ | J ′ ) ∈ E m,n with σ ( I ′ , J ′ ) < σ ( I, J ); and (ii) I is not an interval.4efine i := min( I ), k := max( I ) and A := I − { i, k } . Take ℓ ∈ J and let B := J − ℓ .Since I is not an interval, there is j ∈ [ m ] such that i < j < k and j / ∈ I . Then j / ∈ A and( Aik | Bℓ ) = ( I | J ). Applying to f and g Pl˝ucker-type relations as in (1.5), we have f ( Aj | B ) f ( Aik | Bℓ ) = f ( Aij | Bℓ ) f ( Ak | B ) + f ( Ajk | Bℓ ) f ( Ai | B ) , and (2.1) g ( Aj | B ) g ( Aik | Bℓ ) = g ( Aij | Bℓ ) g ( Ak | B ) + g ( Ajk | Bℓ ) g ( Ai | B ) . (2.2)The choice of i, j, k, ℓ provides that in these relations, the number σ ( A ′ , B ′ ) for each ofthe five corteges ( A ′ | B ′ ) different from ( Aik | Bℓ ) (= ( I | J )) is strictly less than σ ( I | J ). So f and g coincide on these ( A ′ | B ′ ), by condition (i) on ( I | J ). Subtracting (2.2) from (2.1), weobtain f ( Aj | B ) ( f ( I | J ) − g ( I | J )) = 0 . This implies f ( I | J ) = g ( I | J ) (since f ( Aj | B ) = 0 and f ( Aj | B ) is not a zerodivisor, by thehypotheses of the theorem). Case 2 . Let ( I | J ) be subject to condition (i) from the previous case and suppose that J isnot an interval. Then taking i := min( J ), k := max( J ), B := J − { i, k } , ℓ ∈ I , A := I − ℓ ,applying to f, g the corresponding co-Pl˝ucker-type relations as in (1.6), and arguing as above,we again obtain f ( I | J ) = g ( I | J ).Thus, it remains to examine double intervals ( I | J ). We rely on the equalities f ( I | J ) = g ( I | J ) when ( I | J ) is pressed (belongs to P int ), and use induction on the value η ( I, J ) := max( I ) + min( I ) + max( J ) + min( J ) . Case 3 . Let ( I | J ) ∈ E m,n be a non-pressed double interval. Define i := min( I ) − k :=max( I ), j := min( J ) − ℓ := max( J ), A := I − k , B := J − ℓ . Then i, j ≥ I | J ) isnon-pressed). Also ( I | J ) = ( Ak | Bℓ ). Suppose, by induction, that f ( I ′ | J ′ ) = g ( I ′ | J ′ ) holdsfor all double intervals ( I ′ | J ′ ) ∈ E m,n such that η ( I ′ , J ′ ) < η ( I, J ).Applying to f and g Dodgson-type relations as in (1.7), we have f ( Ai | Bj ) f ( Ak | Bℓ ) = f ( Aik | Bjℓ ) f ( A | B ) + qf ( Ai | Bℓ ) f ( Ak | Bj ) , and (2.3) g ( Ai | Bj ) g ( Ak | Bℓ ) = g ( Aik | Bjℓ ) g ( A | B ) + qg ( Ai | Bℓ ) g ( Ak | Bj ) . (2.4)One can see that for all corteges ( A ′ | B ′ ) occurring in these relations, except for ( Ak | Bℓ ),the value η ( A ′ , B ′ ) is strictly less than η ( I, J ). Therefore, subtracting (2.4) from (2.3) andusing induction on η , we obtain f ( Ai | Bj ) ( f ( Ak | Bℓ ) − g ( Ak | Bℓ )) = 0 , whence f ( I | J ) = g ( I | J ), as required.This completes the proof of the theorem. 5 Flows in a planar grid
The proof of Theorem 1.2 essentially relies on a construction of quantum minors via certainpath systems (“flows”) in a special planar graph. This construction is due to Casteels [2]and it was based on ideas in Cauchon [3] and Lindstr˝om [8]. Below we review details of themethod needed to us, mostly following terminology, notation and conventions used for thecorresponding special case in [4].
Extended grids.
Let m, n ∈ Z > . We construct a certain planar directed graph, called an extended m × n grid and denoted as G m,n = G = ( V, E ), as follows.(G1) The vertex set V is formed by the points ( i, j ) in the plane R such that i ∈ { }∪ [ m ], j ∈ { } ∪ [ n ] and ( i, j ) = (0 , i of a point ( i, j ) in the plane is the vertical one.(G2) The edge set E consists of edges of two types: “horizontal” edges, or H-edges , and“vertical” edges, or
V-edges .(G3) The H-edges are directed from left to right and go from ( i, j −
1) to ( i, j ) for all i = 1 , . . . , m and j = 1 , . . . , n .(G4) The V-edges are directed downwards and go from ( i, j ) to ( i − , j ) for all i = 1 , . . . , m and j = 1 , . . . , n .Two subsets of vertices in G are distinguished: the set R = { r , . . . , r m } of sources , where r i := ( i, C = { c , . . . , c n } of sinks , where c j := (0 , j ). The other vertices arecalled inner and the set of these (i.e., [ m ] × [ n ]) is denoted by W = W G .The picture illustrates the extended grid G , . r r r c c c c Each inner vertex v ∈ W of G = G m,n is regarded as a generator . This gives rise toassigning the weight w ( e ) to each edge e = ( u, v ) ∈ E (going from a vertex u to a vertex v )in a way similar to that introduced for Cauchon graphs in [2], namely:(3.1) (i) w ( e ) := v if e is an H-edge with u ∈ R ;(ii) w ( e ) := u − v if e is an H-edge and u, v ∈ W ;(iii) w ( e ) := 1 if e is a V-edge.This in turn gives rise to defining the weight w ( P ) of a directed path P =( v , e , v , . . . , e k , v k ) (where e i is the edge from v i − to v i ) to be the ordered (from left6o right) product, namely: w ( P ) := w ( e ) w ( e ) · · · w ( e k ) . (3.2)Then w ( P ) forms a Laurent monomial in elements of W . Note that when P beginsin R and ends in C , its weight can also be expressed in the following useful form: if u , v , u , v , . . . , u d − , v d − , u d is the sequence of vertices where P makes turns (from “east”to “sought” at each u i , and from “sought” to “east” at each v i ), then, due to the “telescopiceffect” caused by (3.1)(ii), there holds w ( P ) = u v − u v − · · · u d − v − d − u d . (3.3)We assume that the elements of W obey quasi-commutation laws which look somewhatsimpler than those in (1.1); namely, for distinct inner vertices u = ( i, j ) and v = ( i ′ , j ′ ),(3.4) (i) if i = i ′ and j < j ′ , then uv = qvu ;(ii) if i > i ′ and j = j ′ , then vu = quv ;(iii) otherwise uv = vu ,where, as before, q ∈ K ∗ . (Note that G has a horizontal (directed) path from u to v in (i),and a vertical path from u to v in (ii).) Path matrix and flows.
To be consistent with the vertex notation in extended grids, wevisualize matrices in the Cartesian form: for an m × n matrix A = ( a ij ), the row indexes i = 1 , . . . , m are assumed to grow upwards, and the column indexes j = 1 , . . . , n from left toright.Given an extended m × n grid G = G m,n = ( V, E ) with the corresponding partition(
R, C, W ) of V as above, we form the path matrix Path = Path G of G in a spirit of [2];namely, Path is the m × n matrix whose entries are defined byPath( i | j ) := X P ∈ Φ G ( i | j ) w ( P ) , ( i, j ) ∈ [ m ] × [ n ] , (3.5)where Φ G ( i | j ) is the set of (directed) paths from the source r i to the sink c j in G . Thus, theentries of Path G belong to the K -algebra L G of Laurent polynomials generated by the set W if inner vertices of G subject to (3.4). Definition.
Let ( I | J ) ∈ E m,n . Borrowing terminology from [5], by an ( I | J )- flow we meana set φ of pairwise disjoint directed paths from the source set R I := { r i : i ∈ I } to the sinkset C J := { c j : j ∈ J } in G .The set of ( I | J )-flows φ in G is denoted by Φ( I | J ) = Φ G ( I | J ). We order the pathsforming φ by increasing the indexes of sources: if I consists of i (1) < i (2) < · · · < i ( k ) and J consists of j (1) < j (2) < · · · < j ( k ) and if P ℓ denotes the path in φ beginning at r i ( ℓ ) , then P ℓ is just ℓ -th path in φ , ℓ = 1 , . . . , k . Note that the planarity of G and the fact that thepaths in φ are pairwise disjoint imply that each P ℓ ends at the sink c j ( ℓ ) .Similar to the assignment of weights for path systems in [2], we define the weight of φ = ( P , P , . . . , P k ) to be the ordered product w ( φ ) := w ( P ) w ( P ) · · · w ( P k ) . (3.6)Using a version of Lindstr¨om Lemma, Casteels showed a correspondence between pathsystems and q -minors of path matrices. 7 roposition 3.1 ([2]) For the extended grid G = G m,n and any ( I | J ) ∈ E m,n , ∆( I | J ) Path G ,q = X φ ∈ Φ G ( I | J ) w ( φ ) . (3.7)(This is generalized to a larger set of graphs and their path matrices in [4, Theorem 3.1].)The next property, surprisingly provided by (3.4), is of most importance to us. Proposition 3.2 ([2])
The entries of
Path G obey Manin’s relations (similar to thosein (1.1)). It follows that the q -minors of Path G satisfy all universal QIs, and therefore, the function g : E m,n → L G defined by g ( I | J ) := Path G ( I | J ) is a QI-function. Let f : P int m,n → A ∗ be a function as in the hypotheses of this theorem. Our goal isto extend f to a QI-function f on E m,n . The idea of our construction is prompted byPropositions 3.1 and 3.2; namely, we are going to obtain the desired f as the function of q -minors of an appropriate path matrix Path G for the extended m × n grid G = G m,n .For this purpose, we first have to determine the “generators” in W in terms of values of f (so as to provide that these values are consistent with the corresponding pressed interval q -minors of the path matrix), and second, using the quasi-commutation relations (as in (1.8))on the values of f , to verify validity of relations (3.4) on the generators. Then P ath G willbe indeed a fine q -matrix and its q -minors will give the desired QI-function f .(It should be emphasized that we may speak of a vertex of G in two ways: either as apoint in R , or as a generator of the corresponding algebra. In the former case, we use thecoordinate notation ( i, j ) (where i ∈ { } ∪ [ m ] and j ∈ { } ∪ [ n ]). And in the latter case,we use notation w ( i, j ), referring to it as the weight of ( i, j ).)To express the elements of W via values of f , we associate each pair ( i, j ) ∈ [ m ] × [ n ]with the pressed interval cortege π ( i, j ) = ( I | J ), where(4.1) I := [ i − k + 1 ..i ] and J := [ j − k + 1 ..j ], where k := min { i, j } .In other words, if i ≤ j (i.e., ( i, j ) lies “south-east” from the “diagonal” { α, α } in R ),then ( I | J ) is the co-flag interval cortege with I = [ i ] and max( J ) = j , and if i ≥ j (i.e., ( i, j )is “north-west” from the diagonal), then ( I | J ) is the flag interval cortege with max( I ) = i and J = [ j ]. Also it is useful to associate to ( i, j ): the (almost rectangular) subgrid inducedby the vertices in ( { } ∪ [ i ]) × ( { } ∪ [ j ]) − { (0 , } , and the diagonal D ( i | j ) formed by thevertices ( i, j ) , ( i − , j − , . . . , ( i − k + 1 , j − k + 1). See the picture where the left (right)fragment illustrates the case i < j (resp. i > j ), the subgrids are indicated by thick lines,and the diagonals D ( i, j ) by bold circles. 8 r i c c j r r i c c j (i,j) (i,j) An important feature of a pressed interval cortege ( I | J ) ∈ E m,n (which is easy to see) isthat(4.2) Φ( I | J ) consists of a unique flow φ and this flow is formed by paths P , . . . , P k , wherefor i := max( I ), j := max( J ), k := min { i, j } , and ℓ = 1 , . . . , k , the path P ℓ begins at r i − k + ℓ , ends at c j − k + ℓ and makes exactly one turn, namely, the east to sought turn atthe vertex ( i − k + ℓ, j − k + ℓ ) of the diagonal D ( i | j ).We denote this flow ( P , . . . , P k ) as φ ( i | j ); it is illustrated in the picture (for both cases i < j and i > j from the previous picture). r r i c c j r r i c c j (i,j) P k P (i,j) P k P Therefore, for each ( i, j ) ∈ [ m ] × [ n ], taking the cortege ( I | J ) = π ( i, j ) and the flow φ ( i | j ) = ( P , . . . , P k ) with k = min { i, j } and using expressions (3.3) and (3.6) for them, weobtain that X φ ∈ Φ G ( I | J ) w ( φ ) = w ( φ ( i | j )) = w ( i − k + 1 , j − k + 1) · · · w ( i − , j − w ( i, j ) . (4.3)Now imposing the conditions w ( φ ( i | j )) := f ( I | J ) for all ( I | J ) = π ( i, j ) ∈ P int m,n , (4.4)we come to the rule of defining appropriate weights of inner vertices of G . Namely, relyingon (4.3), we define w ( i, j ) for each ( i, j ) ∈ [ m ] × [ n ] by w ( i, j ) := (cid:26) f ( { i }|{ j } ) if min { i, j } = 1 , ( f ( π ( i − , j − − f ( π ( i, j )) otherwise . (4.5)Such a w ( i, j ) is well-defined since f ( π ( i − , j − i, j ) and ( i ′ , j ′ ), 94.6) (i) if i = i ′ and j < j ′ , then w ( i, j ) w ( i ′ , j ′ ) = qw ( i ′ , j ′ ) w ( i, j );(ii) if i > i ′ and j = j ′ , then w ( i ′ , j ′ ) w ( i, j ) = qw ( i, j ) w ( i ′ , j ′ );(iii) otherwize w ( i, j ) w ( i ′ , j ′ ) = w ( i ′ , j ′ ) w ( i, j ).This would provide that P ath G is indeed a fine q -matrix, due to (3.7) and Proposition 3.2,and setting f ( I | J ) := ∆( I | J ) P ath G for all ( I | J ) ∈ E m,n , we would obtain the desired function,thus completing the proof of the theorem.First of all we have to explain that(4.7) f satisfies the quasi-commutation relation for any two pressed interval corteges( I | J ) , ( I ′ | J ′ ) ∈ P int , i.e., f ( I | J ) f ( I ′ | J ′ ) = q c f ( I ′ | J ′ ) f ( I | J ) holds for some c ∈ Z .This is equivalent to saying that such corteges determine a universal QI of the form (1.8)on associated q -minors. To see the latter, assume that | I | ≥ | I ′ | and define α, β, γ, δ asin (1.9). One can check that: γ = δ = 0 if both interval corteges are flag ones; α = β = 0 ifthey are co-flag ones; and either β = γ = 0 or α = δ = 0 (or both) if one of these is a flag,and the other a co-flag interval cortege. So in all cases, we have αγ = βδ = 0, and (4.7)follows from (1.10).Next we start proving (4.6). Given ( i, j ) , ( i ′ , j ′ ) ∈ [ m ] × [ n ], let ( I | J ) := π ( i, j ) and( I ′ | J ′ ) := π ( i ′ , j ′ ), and define A := f ( I | J ) , B := f ( I − i | J − j ) , C := f ( I ′ | J ′ ) , D := f ( I ′ − i ′ | J ′ − j ′ ) , letting by definition B := 1 ( D := 1) if | I | = 1 (resp. | I ′ | = 1). (Here for an element p ∈ P ,we write P − p for P − { p } .)Then w ( i, j ) is rewritten as B − A , and w ( i ′ , j ′ ) as D − C (by (4.5)), and our goal is toshow that B − AD − C = q d D − CB − A, (4.8)where d is as required in (4.6) (i.e., equal to 1, -1, 0 in cases (i),(ii),(iii), respectively).Define c , c , c , c from the quasi-commutation relations (as in (1.8)) AC = q c CA, AD = q c DA, BC = q c CB, BD = q c DB. (4.9)One can see that d = c − c − c + c . (4.10)Indeed, in order to transform the string B − AD − C into D − CB − A , one should swap eachof A, B − with each of C, D − . The second equality in (4.9) implies AD − = q − c D − A , andfor similar reasons, B − C = q − c CB − and B − D − = q c D − B − .Now we are ready to examine possible combinations for ( i, j ) and ( i ′ , j ′ ) and compute d in these cases by using (4.10). We will denote the intervals I − i, J − j, I ′ − i ′ , J ′ − j ′ in question by e I, e J , e I ′ , e J ′ , respectively. Also for an ordered pair (( P | Q ) , ( P ′ | Q ′ )) of doubleintervals in E m,n (where | P ′ | = | Q ′ | may exceed | P | = | Q | ), we define α ( P, P ′ ) := min {|{ p ′ ∈ P ′ : p ′ < min( P ) }| , |{ p ∈ P : p > max( P ′ ) }|} ; β ( P, P ′ ) := min {|{ p ′ ∈ P ′ : p ′ > max( P ) }| , |{ p ∈ P : p < min( P ′ ) }|} , (4.11)10nd define γ ( Q, Q ′ ) and δ ( Q, Q ′ ) in a similar way (this matches the definition of α, β, γ, δ in (1.9) when | P | ≥ | P ′ | ). Using (1.10), we observe that the sum β ( I, I ′ )+ δ ( J, J ′ ) − α ( I, I ′ ) − γ ( J, J ′ ) is equal to c , and similarly for the pairs concerning c , c , c .In our analysis we also will use the values ϕ := ( β ( I, I ′ ) − α ( I, I ′ )) − ( β ( I, e I ′ ) − α ( I, e I ′ )) − ( β ( e I, I ′ ) − α ( e I, I ′ )) + ( β ( e I, e I ′ ) − α ( e I, e I ′ )); ψ := ( δ ( J, J ′ ) − γ ( J, J ′ )) − ( δ ( J, e J ′ ) − γ ( J e J ′ )) − ( δ ( e J, J ′ ) − γ ( e J , J ′ )) + ( δ ( e J , e J ′ ) − γ ( e J , e J ′ )) . In view of (4.10) and (4.11), ϕ + ψ = c − c − c + c = d. (4.12)The lemmas below compute ϕ using (4.11). Let r := min( I ) (= min( e I )) and r ′ := min( I ′ )(= min( e I ′ )). Lemma 4.1
Suppose that | I | 6 = | I ′ | and i = i ′ . Then ϕ = 0 . Proof
Assume that | I | > | I ′ | . Then | I | > | e I | ≥ | I ′ | > | e I ′ | . Consider possible cases. Case 1 : r ≤ r ′ and i ′ < i . Then I ′ , e I ′ ⊆ I, e I . Therefore, both α and β are zero everywhere,implying ϕ = 0. Case 2 : I ∩ I ′ = ∅ . If i ′ < r , then β is zero. Also α ( I, I ′ ) = | I ′ | = α ( e I, I ′ ) and α ( I, e I ′ ) = | e I ′ | = α ( e I, e I ′ ).And if i < r ′ , then α is zero. Also β ( I, I ′ ) = | I ′ | = β ( e I, I ′ ) and β ( I, e I ′ ) = | e I ′ | = β ( e I, e I ′ ).So in both situations, ϕ = 0. Case 3 : r ′ < r ≤ i ′ < i . Then β is zero. Also α ( P, P ′ ) = r − r ′ holds for all P ∈ { I, e I ) and P ′ ∈ { I ′ , e I ′ } , implying ϕ = 0. Case 4 : r < r ′ ≤ i < i ′ . Then α is zero, and β ( I, I ′ ) = i ′ − i = β ( e I, e I ′ ) , β ( I, e I ′ ) = i ′ − − i and β ( e I, I ′ ) = i ′ − ( i − , again implying ϕ = 0.When | I | < | I ′ | , the argument follows by swapping I, e I by I ′ , e I ′ . Lemma 4.2
Let | I | = | I ′ | . (a) If i < i ′ then ϕ = 1 . (b) If i > i ′ then ϕ = − . (c) If i = i ′ then ϕ = 0 . Proof
We have | I ′ | , | e I ′ | ≤ | I | and | e I ′ | = | e I | but | I ′ | = | e I | +1. Let i > i ′ . Then, using (4.11)),one can check that β is zero. Also if I ∩ I ′ = ∅ , then α ( I, I ′ ) = | I | , α ( I, e I ′ ) = | e I ′ | = α ( e I, e I ′ ) , α ( e I, I ′ ) = | e I | = | I | − . And if I ∩ I ′ = ∅ , then α ( I, I ′ ) = α ( I, e I ′ ) = α ( e I, e I ′ ) = r − r ′ = i − i ′ and α ( e I, I ′ ) = | e I − I ′ | = ( i − − i ′ . ϕ = − α ( I, I ′ ) + α ( I, e I ′ ) + α ( e I, I ′ ) − α ( e I, e I ′ ) = α ( e I, I ′ ) − α ( I, I ′ ) = − , as required in (b).Case (a) reduces to (b). And if i = i ′ then r = r ′ , implying that both α, β are zero (sincefor any two intervals among I, e I, I ′ , e I ′ , one is included in the other). Lemma 4.3
Let i = i ′ . (a) If | I | > | I ′ | then ϕ = − . (b) If | I | < | I ′ | then ϕ = 1 . Proof
Let | I | > | I ′ | . Then I ′ , e I ⊂ I and e I ′ ⊂ e I . Hence α and β are zero on each of( I | I ′ ) , ( I | e I ′ ) , ( e I | e I ′ ). Also | e I | ≥ | I ′ | and r < r ′ imply α ( e I, I ′ ) = 0 and β ( e I, I ′ ) = i ′ − ( i −
1) = 1(since max( e I ) = i − ϕ = − β ( e I, I ′ ) = − i, i ′ by j, j ′ , and I, I ′ by J, J ′ in Lemmas 4.1–4.3, we obtain the correspondingstatements concerning ψ .(4.13) (i) If | J | = | J ′ | and j < j ′ , or if | J | < | J ′ | and j = j ′ , then ψ = 1.(ii) Symmetrically, if | J | = | J ′ | and j > j ′ , or if | J | > | J ′ | and j = j ′ , then ψ = − ψ = 0.Now we finish the proof with showing (4.6) in the corresponding three cases. Case A : i = i ′ and j < j ′ . First suppose that i ≤ j . Then both ( I | J ) and ( I ′ | J ′ ) are co-flagcorteges, and | I | = | I ′ | = i . We have ϕ = 0 (by Lemma 4.2(c)) and ψ = 1 (by (4.13)(i)).Next suppose that j < i < j ′ . Then ( I | J ) is flag, ( I ′ | J ′ ) is co-flag, and | I | = j < i = | I ′ | .This gives ϕ = 1 (by Lemma 4.3(b)) and ψ = 0 (by (4.13)(iii)).Finally, suppose that j ′ ≤ i . Then both ( I | J ) , ( I ′ | J ′ ) are flag, and | I | = j < j ′ = | I ′ | .This gives ϕ = 1 (by Lemma 4.3(b)) and ψ = 0 (by (4.13)(iii)).Thus, in all situations, d = ϕ + ψ = 1, as required in (4.6)(i). Case B : i < i ′ and j = j ′ . This is symmetric to the previous case, yielding d = 1. Thismatches assertion (ii) in (4.6) (since replacing i < i ′ by i > i ′ changes d = 1 to d = − Case C : i = i and j = j ′ . When ϕ = ψ = 0, (4.6)(iii) is immediate. The situation with ϕ = 0 arises only when | I | = | I ′ | ; then (a) i < i ′ implies ϕ = 1, and (b) i > i ′ does ϕ = − ψ = 0 happens only if | J | = | J ′ | ; then (c) j < j ′ implies ψ = 1,and (d) j > j ′ does ψ = − i < i ′ and | I | = | I ′ | =: k imply i ′ > k (in view of i ≥ | I | ). Therefore, j ′ = k must hold (i.e., ( I ′ | J ′ ) is flag). Then j = j ′ implies j > j ′ , and we obtain ψ = − i > i ′ and | I | = | I ′ | =: k imply i > k . Therefore, j = k . Then j ′ > j ,yielding ψ = 1, by (4.13)(i).So in both (a) and (b), we obtain ϕ + ψ = 0. In their turn, subcases (c) and (d) aresymmetric to (a) and (b), respectively. Thus, in all situations, d = 0 takes place, as requiredin (4.6)(iii).This completes the proof of Theorem 1.2.12 Uniqueness
Let f : P int m,n → A ∗ be a function in the hypotheses of Theorem 1.2, i.e., f satisfies quasi-commutation relations for all pairs of pressed interval corteges in E m,n (cf. (4.7)). A priori, f may have many extensions to E m,n that are QI-functions. One of them is the function f whose values f ( I | J ) are q -minors ∆( I | J ) of the corresponding path matrix constructed inthe proof in Sect. 4.In light of Theorems 1.1 and 1.2, it is tempting to ask when f has a unique QI-extension.Since any QI-extension is an RQI-function (i.e., satisfies the corresponding relations ofPl˝ucker, co-Pl˝ucker and Dodgson types) and in view of Theorem 1.1, we may address anequivalent question: when an RQI-extension g of f is a QI-function (and therefore g = f ).We give sufficient conditions below (which is, in fact, a corollary of Theorems 1.1 and 1.2).To this aim, let us associate to each ( I | J ) ∈ P int m,n an indeterminate y I | J and formthe K -algebra L Y of quantized Laurent polynomials generated by these y I | J (where thequantization is agreeable with that for f ). The values of f are said to be algebraicallyindependent if the map y I | J f ( I | J ), ( I | J ) ∈ P int m,n , gives an isomorphism between L Y and the K -subalgebra A f of A generated by these values. Corollary 5.1
Let f and f be as above. Let the following additional conditions hold: (i) the values of f are algebraically independent; (ii) if an element a ∈ A f is a zerodivisor in A , then a is a zerodivisor in A f .Suppose that g is an RQI-function on E m,n coinciding with f on P int m,n . Then g is aQI-function (and therefore g = f ). Proof (a sketch) Considering the construction of q -minors of the path matrix related to f (cf. (3.5),(3.7),(4.3)–(4.5)), one can deduce that for each cortege ( I | J ) ∈ E m,n , y I | J is anonzero polynomial in L Y . Then condition (i) implies that f ( I | J ) is a nonzero element of A f . Furthermore, since L Y is free of zerodivisors (by a known fact on Laurent polynomials;see, e.g. [1], ch. II, $11.4, Prop. 8), so is A f . Therefore, by condition (ii), f ( I | J ) is not azerodivisor in A . Now applying Theorem 1.1, we obtain g = f , as required. References [1] N. Bourbaki, Algebre, Hermann, Paris, 2007.[2] K. Casteels, A graph theoretic method for determining generating sets of prime ideals inquantum matrices,
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