Proc. Natl. Acad. Sci. USA Vol. 88, pp. 8087-8090, September 1991 Higher-order syzygies for the and for the 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 over a signed alphabet. As a a2, . . ., aj are letters, we 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 . 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 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,. , w')Tab(w'+,.., Wk)

Wr- 1 Vr) with nonzero integer coefficients f8, where each ofthe Young is standard; diagrams (w', W2,. ., wO) is antistandard, and where each 3. SYZ(W1, W2, . . ., Wr-1)[Wrl of the Young diagrams (w., W., . w..,L) is standard. We prove simultaneously Theorem 2 and Theorem 3, = E 6iSyz(vl, V2, . _,v1)[vr]; and proceeding by induction on r. Let p be an element in the of dri . We may write p 4. each of the Young diagrams in the form

(v, V,... Vr) p = 2 'yj(w4, 2wi. ..., wi_1)Tab(wi, ... .9 W), has strictly lower lexicographic order than the Young dia- gram (W1, W2, . . .,wr) where 'yj is an integer and where, by Proposition 1, we may The integers 6i and the Young diagrams assume that each of the Young diagrams (wi, wj,, . . ., wI) is standard.

(v , V2, . . .*, Vr 1, Vi) Our assumption is that Downloaded by guest on September 30, 2021 Mathematics: Anick and Rota Proc. Natl. Acad. Sci. USA 88 (1991) 8089 We proceed by induction on the order of p. We are dr-lP = dr-(Iy(w, wi2, *., W1r-l)Tab(wr.**, Wik)) assuming that the order ofp equals the lexicographic order of the diagram (wj, WI, Wi,** ., wi). We may write p in the form = > yiSyz(wjl, w ,..., Wjr-l)Tab(wjr, *wk) = 0. P = Yi(wi, w2, . . ., wl.1)Tab(w1, . . ., Wk) + p", Now, if all Young diagrams (wr1, wr.w, ) were standard, then if r = 2, then all integers yj would be equal to where either p' = 0 or p" has strictly lower order than p, and 0, by Theorem 1, and we infer that p = 0, and if instead r > where yi # 0. 2, suppose that yi # 0, and suppose that (wi, Wi, W4..., wI) Let the integers 8A and the standard Young diagrams (v'I, v2, is lexicographically the largest among all Young diagrams vI) (where vb vI,.. ., vI are words of length n) be the (wj, w2, w3, . . ., w) with yj # 0. Since straightening data of order r for the antistandard Young diagram (w', w', wri, wl). By the induction hypothesis 0= dr-1p = ylSyz(wl, W2,.., w_)Tab(wr,.. ., Wk) applied to Theorem 2, we may assume the identity + > 'y1Syz(w4, wi2, . .., wi_1)Tab(wj, *.,w..)9 Syz(wl, W,9 w1 )[w1] = I 8sSyz(V*V.-. V] joi and since so that

y1Syz(w, w, .. ., w 1)Tab(w ,..., w1) = Syz(wl, wig ...* * W1) 1(WlW2,r3,* *k*,1 1 =(1 s W2, Wr-l[r- 6svg . . . 9 V-[Vr]. Viwi 2~ 3~ . w2)Tab(wr..19r Wr, w3 , ~k +p, Aw( where p' is a linear combination with integer coefficients of We therefore have elements of Young(r - 2) of grade k, each of which has 1 strictly smaller lexicographic order than (wj, WI, W4 ... = yi(wl, wg .. , )[wTab(wl+l, w+2 w) + p W1), it follows that - yISyz(wI, W2,..., wr)Tab(w +1, ..., Wk) + p" 'yj(W, W 1, W3 . , Wr-)a(l- s W3,... W") - V2 65Syz(v, ..., vss.)Tab(vs, Wi+i ...w) S = -p2 3 bSyz(w'lr w r,.1.., Wir)Tab(w,.*, joi - vSyz(wl, w2,. . ., w')Tab(wl+l, w1+2, . . ., Wk) + p"', But where p"' has strictly lower order than p. E jSyz(wjg wig ... ., wi_j)Tab(wig .. ., Wi) We have thus shown that is also a linear combination with integer coefficients of P = ylsyz(wl,w9)Tab(wl~l,...,wi,..., wk) + 9r" elements of Young(r - 2) of grade k, each of which has where strictly smaller lexicographic order than (wj, wI, WI, .. ., Wk). Hence, by Proposition 2, we must have yi = 0, a contradic- dr-i(P') tion. We again conclude that all integers yj equal 0, and hence that p = 0. Thus, in both cases, ifp # 0, there exists aj such = dr- (p -ySyz(wi, w2, ..., wl)Tab(wl+i, ..., wi)) that the Young diagram (wr-i1, w{,. . ., wI) is not standard; we may and shall assume that the highest lexicographic order - dr-i(p)- dr,(ySyz(wl, w2, .. ., w)Tab(wl+1, ... ., wi)) among all Young diagrams (wi, wi, wi,..., w4) such that (wJr-1, wj, wI) is not standard is attained for the Young =0. diagram (wj, WI,w1 , . . ., W . To summarize, the Young diagram (w',w., W3, . . ., Wk) has the following properties: Furthermore, p' has strictly lower order than p. Therefore, 1. The Young diagram (w, w . ., w) is standard. induction on the order ofp can be carried out, to prove that 2. The Young diagram (w~l~, w,..., wI) is not standard. every element of grade k in the kernel of dri1 is a linear 3. The Young diagram (Wlil, wi) is antistandard. combination with integer coefficients of elements of the form 4. The Young diagram (wj, WI, Wi,.. ., W~I ) is antistan- dard. Syz(wi, Wig ... ., w')Tab(w'+,.. ., Wi) 5 (combining 3 and 4). The Young diagram (WI, W2, W3, w1) is antistandard. where each of the Young diagrams (wi, wi, ..., w,) is 6. other antistandard, and where each of the Young diagrams (w,, Any Young diagram (wj, w4, . wr_1, w.. w,+l . . ., is standard. Wj) such that (w'ri, w{, . .,) is not standard, and which wO appears with a nonzero integer coefficient yi, has lower It remains to be shown that such a linear combination is lexicographic order than the Young diagram (w,1w, W,. .. unique. Again, we prove this assertion by induction on order. wl). Thus, suppose that For purposes of the present proof, let the order ofp be the = 09 8 Wk) highest lexicographic order of any Young diagram (w{l, wj2, iSyz(w'1, w',..., wr)Tab(w'+i,..., [21 w .,wjO such that (w{-1, w{,., wj) is not standard, appearing with a nonzero coefficient in where the Young diagrams (wj, w.., . ., w,) are antistandard yj and where the Young diagrams (w', wr+i, .. ., w) are standard. We may assume that 13i # 0, and that the Young IE (l, W 2 ., w-r- )Tab(wJr * wik) diagram (wj, w, ..., wk) has the highest lexicographic order Downloaded by guest on September 30, 2021 8090 Mathematics: Anick and Rota Proc. Natl. Acad. Sci. USA 88 (1991) among all Young diagrams (wi, w , . . ., w') such that P3i # They are 0. Thus, we can rewrite Eq. 2 in the form (ad, bc, a), (bd, bc, a), (cd, bc, a), (cd, bd, a), Wig ... .s wl)Tab(wl~l . , PjlSyz(W1' and =-1 1 , w2, . . ., . w . [3] bi2ri#...1kf3iSyz(w w)Tab(w' d(ad, bc, a) = [ad](bc, a) - [ac](bd, a) + [ab](cd, a) By the induction hypothesis of Theorem 2 we have, for every d(bd, bc, a) = [bd](bc, a) - [bc](bd, a) + [ab](cd, b) i, d(cd, bc, a) = [cd](bc, a) - [bc](cd, a) + [ac](cd, b) Syz(w', w2,. , . ,w ) wrl)Tab(wr'+,, d(cd, bd, a) = [cd](bd, a) - [bd](cd, a) + [ad](cd, b). = (wi, w2, .. ., w 1)Tab(w', . . *, w') + , 3. Syzygies of order 4. where pi is a linear combination with integer coefficients of They are homogeneous terms of grade k, each of which is of strictly lower lexicographic order than the Young diagram (wi, wi, (bc, ad, bc, a), (bd, ad, bc, a), (cd, ad, bc, a), Wi). Thus, Eq. 3 can be rewritten as (cd, bd, bc, a), 381(wl, w2,) .. ., Wr_ )Tab(wr,..* Wk) =-lPi and - (,8i((w',1 w', . . ., w'-1)Tab(w'r,rk.. ., W) + pi). [4] i#1 d(bc, ad, bc, a) = [bc](ad, bc, a) - [ac](bd, bc, a) But every term occurring on the right side of Eq. 4 is of + [ab](cd, bc, a) strictly lower lexicographic order than (wi, wif .. ., wk). Thus, 81 = 0, a contradiction by Proposition 2. d(bd, ad, bc, a) = [bd](ad, bc, a) - [ad](bd, bc, a) This completes the proofs of Theorems 2 and 3. As an immediate consequence of Theorem 3 we have + [ab](cd, bd, a) THEOREM 4. The sequence of modules Young(r) and op- erators dr is exact, that is, image(dr) equals kernel(dri-). d(cd, ad, bc, a) = [cd](ad, bc, a) - [ad](cd, bc, a) We thus obtain a bijective correspondence between anti- standard Young diagrams (w1, w2, .. ., wr) and basic syzygies + [ac](cd, bd, a) of order r given by the operator dr, to wit: d(cd, bd, bc, a) = [cd](bd, bc, a) - [bd](cd, bc, a) dr: (W1, W2, .. ., Wr) - Syz(wl, W2, Wr)- + [bc](cd, bd, a) 5. Section Examples Observe that the for syzygies of order 4 coincides with the matrix of syzygies of order 2. It can be shown that We give the syzygies of various order for the classical case the syzygy matrix of syzygies of order 5 coincides with that of a bracket of length n = 2, over a negative alphabet. These for syzygies of order 3, etc. brackets occur in the symbolic method for the of We conjecture that such periodicity of syzygy matrices is invariants of binary forms. For ease of notation, we have the rule for all supersymmetric brackets of any order. inverted the order of product of the brackets. 1. Syzygies of order 2. The authors thank Henry Crapo for contributing the computation They are of syzygies in Section 5. The work of G.-C.R. was supported by National Science Foundation Grant MCS 81 D4855 and D.A.'s work (bc, a), (bd, a), (cd, a), (cd, b), was supported by National Science Foundation Grant 8912842. 1. Study, E. (1889) Methoden der Theorie der ternare Formen and (Springer, Berlin) reprinted (1982) by Springer, Berlin. 2. Anick, D. (1986) Trans. Am. Math. Soc. 296 (2), 641-659. d(bc, a) = [bc]a - [ac]b + [ab]c 3. White, N. (1975) Trans. Am. Math. Soc. 202, 79-103. 4. Dress, A. & Wentzel, W. (1991) Adv. Math. 86, 68-110. d(bd, a) = [bd]a - [ad]b + [ab]c 5. Grosshans, F., Rota, G.-C. & Stein, J. (1986) Invariant Theory and Superalgebras (Am. Math. Soc., Providence, RI). d(cd, a) = [cd]a - [ad]c + [ac]d 6. Huang, R., Rota, G.-C. & Stein, J. (1989) Symmetry in Nature, ed. Vesentine, E. (Scuola Normale Superiore, Pisa, Italy), Vol. d(cd, b) = [cd]b - [bd]c + [bc]d. 2, pp. 407-432. 7. Huang, R., Rota, G.-C. & Stein, J. A. (1990) Acta Applicandae 2. Syzygies of order 3. Math. 21, 193-246. Downloaded by guest on September 30, 2021