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Higher-Order Syzygies for the Bracket Algebra and for the Ring Of Proc. Natl. Acad. Sci. USA Vol. 88, pp. 8087-8090, September 1991 Mathematics Higher-order syzygies for the bracket algebra and for the ring of coordinates of the Grassmanian DAVID ANICK AND GIAN-CARLO ROTA Massachusetts Institute of Technology, Cambridge, MA 02139 Contributed by Gian-Carlo Rota, May 8, 1991 ABSTRACT A Poincar6 resolution is given for the super- lexicographic order on words. If w = aja2, . aj, where al, symmetric ring of brackets over a signed alphabet. As a a2, . ., aj are letters, we set length(w) =j. The multiplicative consequence, a resolution is found for the ring of coordinates identity element of Mon(L) will be said to be a word oflength ofthe Grassmanian variety in projective space over any infinite 0. A word w = aja2 .. aj is said to be standard if the Young field. diagram (w) is standard, in the sense of ref. 6 or ref. 7. A Young diagram ofgrade k, denoted by (w1, w2, . .., WJ, is an ordered sequence of standard words wi. We shall assume Section 1. Introduction throughout that in every Young diagram (wl, w2, Wk) we have length(w,) = n for 1 < i s k. The problem of computing the higher-order syzygies for the A 'Young diagram (wl, w2, . ., Wk) will be said to be algebra ofbrackets, or, what is equivalent, for the coordinate antistandard when none of the diagrams (wi, wi+1) are stan- ring of the Grassmanian variety in projective space, was first dard for 1 < i < k - 1. stated by Study (ref. 1, p. 82), in the following words: "Es Young diagrams of the same grade are linearly ordered ware von Interesse, in einigen Beispielen die Natur der lexicographically. We set (w1, w2, ..., Wk) c (Wl, w2, analytischen Gebilde zu untersuchen, die entstehen, wenn wk) whenever w1w2 ... Wk < WlW2 ... wk in the lexico- man in den Identitaten zwischen Invarianten diese letzteren graphic order on words. durch unabhangige Grossen ersetzt." [It would be of interest All rings and algebras will be taken coeffi- to investigate in a few examples the nature of the analytical with integer systems that arise when one substitutes in the identities cients. We consider the bracket algebra Bra[L, n] of rank n between invariants these latter through independent quanti- over a proper alphabet L, as defined in ref. 7. We follow the ties.] Study also gave the first example of a higher-order notation in ref. 7, writing a bracket in the form [w], where w syzygy. We outline a characteristic-free solution of the is a word in Mon(L) such that length(w) = n. problem. The present method is an application ofthe general If (w1, w2,..., Wk) is a Young diagram, we denote by theory of resolutions developed in ref. 2. Because of the added strength that results from the availability of the Tab(wl, W2, * * ., Wk) straightening algorithm for the algebra ofbrackets, the results in ref. 2 can be sharpened in the case we treat here. The the element of Bra[L, n] defined as following presentation does not require background in ho- mological algebra. Tab(wl, W2, . *, Wk) = [W1][W2] . [Wk]j The motivation for the computation of a syzygy chain is to For any nonnegative integer r, we denote by Young(r) the be found not only in the symbolic method of classical free right module over Bra[L, n] generated by all antistandard invariant theory but also in the theory of matroids. The Young . bracket ring of a matroid, defined by White (3), suggests that diagrams (wl, w2, . ., we). Thus, an element of classes of matroids defined by excluded minors may be Young(r) is a linear combination with integer coefficients of alternatively characterized by syzygetic properties of their elements of the form (w1, w2, . ., wr)Tab(wr+l, Wr+2, * * ., bracket ring. The class of binary matroids is easily charac- Wk), where (w1, w2, ..., wr) is an antistandard Young terized in this way, but other notable classes, beginning with diagram, and where (Wr+l, Wr+2, ..., Wk) is any Young the class of regular (unimodular) matroids, have so far diagram. For r = 0, we set Young(O) = Bra[L, n]. An element resisted such treatment. We conjecture that the translation of p of Young(r) that is a linear combination with integer an excluded minor characterization into a property of the coefficients of elements of the form bracket ring can be carried out by use of higher-order syzygies, such as are computed below. Recent work ofDress (W19 W29 ..* . wr)Tab(Wr+l, Wr+29 . sWk) and his school supports this conjecture (ref. 4 and references therein). will be said to be homogeneous ofgrade k. The following is the main property of the bracket algebra Section 2. Notation Bra[L, n] that is used below (see ref. 6 or ref. 7 for the proof): THEOREM 1. (standard basis theorem). Let (w1, w2, We shall use the notation and results of refs. 5-7. Let L be wk) be a Young diagram such that a proper signed alphabet (that is, a linearly ordered set) and let n be a positive integer, which will remain fixed through- Tab(wl, W2, * *., Wk) # 0. out. We denote as in ref. 5 by Mon(L) the free monoid generated by the alphabet L. An element of Mon(L) will be Then there exists a unique finite set {(w', wi, * * ., w0} of called a word. The set Mon(L) is linearly ordered by the standard Young diagrams and a nonzero integer a, associ- ated with each Young diagram (wi, w, . ., wI) belonging The publication costs of this article were defrayed in part by page charge to this set, such that payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact. Tab(wl, W2, . ., wk) = Jj a1Tab(w4, w *, ..).w [1 8087 Downloaded by guest on September 30, 2021 8088 Mathematics: Anick and Rota Proc. Natl. Acad. Sci. USA 88 (1991) Furthermore, if (w1, w2, ...., wk) is not a standard Young will be called the straightening data of order r for the diagram, then for all diagrams (wi, W2,.. ., 4) appearing antistandard Young diagram (wl, w2, * . -, wri1, wr). on the right side ofEq. I we have The proof of Theorem 2 is given together with that of Theorem 3 below. (W1, W2, . WO) < (W19 W2, * * Wk)- We define The set of Young diagrams (wi, wi,. ., Wk) and integers Syz(wl, W2, . ., Wr) = (W1, W2, ** ., Wr-l)[wr] ai will be called the straightening data of the Young diagram (W1, W2, . * ., Wk). - j(V, v, * * ., V)[Vr As a consequence of the standard basis theorem, we have the following. thereby completing the inductive definition of Syz(w1, w2, PROPOSITION 1. The Bra[L, n]-module Young(r) is freely ** wr). We stress the fact that the element Syz(wl, w2, . .. generated as an Abelian group by all elements wr) belongs to the module Young(r -1), not to the module (w1, W2, . ., wr)Tab(wr+l, wr+2, . wk), k - r, Young(r). The element Syz(wl, w2,..., wr) will be called a syzygy of where (w1, W2, .. ., wr) is an antistandard Young diagram, order r. For r 2 1, we define a Bra[L, n]-module homomor- and where (wr+i, Wr+2, ..., Wk) is a standard Young dia- phism dr mapping Young(r) into Young(r - 1), as follows. For gram. r = 1, we set d1(w1) = [w1]; extend the definition of di to a As a consequence of Proposition I and of Theorem 1, we Bra[L, n]-module homomorphism, by setting have the following. PROPOSITION 2. Suppose that Yi is an integer such that dl((w1)Tab(w2, . ., w/O)) = Tab(wl, W2, .,wk) . , . ., wk) + p', and finally extending by linearity over Z. 0=yA(wi, w1, wl)Tab(wl+1, For r > 1, set where the Young diagram (wi+1, . ., wt) is standard, and where the element p' is a linear combination with integer dr(wl, w2, . ., Wr) = Syz(wl, W2, W.); coefficients ofelements ofYoung(r) ofgrade k, each ofwhich has strictly smaller lexicographic order than the Young extend the definition of dr to a Bra[L, n]-module homomor- diagram (wi, wi, .. ., w , Wl+1. .., wi). Then )yi = 0. phism, by setting Section 3. The Syzygies dl((wl, w2, . ., wr)Tab(wr+l, * * ,w = w2, ... , Wk), We inductively define for r 2 1 an element Syz(wl, wr)Tab(wr+1, *I*, over Syz(wl, W2, ** ., Wr) and finally extending by linearity the ring of integers. PROPOSITION 3. of the module Young(r - 1), as follows. dr-ldr = 0. For r = 1, set Syz(w1) = [w1l]. Proof: dr-ldr(wi, w2, . ., wr) = dr-lSyz(wl, w2, ..*, Wpr) The induction to r > 1 is made possible by the following. = dr-l((Wl, w2, . , Wr..)[wrj - Xi SvO, V . ., )Vr Let r = - THEOREM 2 (standard basis theorem oforder r). .2, SYZ(W1, W2 . * ., Wr-..)[wrj Xi siSyz(v'1, V2 * Vr- VA and let the induction hypothesis be that Syz(w1, W2,. = 0. wr-1) is defined. Let (w1, W2, ..., wr) be an antistandard The conclusion follows by linearity. Young diagram. Then there exist a unique finite set of nonzero integers 6i and a uniquefinite set ofYoung diagrams Section 4. Exactness (vj, V2, . .., Vr) such that 1. each of the diagrams THEOREM 3. Let k 2 r 2 2. Every homogenous element of Wig, V2,., Vr'-1) grade k in Ker(dr-1) can be uniquely written as a linear combination is antistandard; 2. each of the diagrams z .iSyz(wi, W2,.
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