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Proc. Nati. Acad. Sci. USA Vol. 87, pp. 653-657, January 1990 Supersymmetric Hilbert (supersymmetric / with straightening laws/Young tableaux/invariant theory, skew-symmetric ) GIAN-CARLO ROTAt AND JOEL A. STEIN: tDepartment of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139; and tDepartment of Mathematics and Statistics, California State University, Chico, CA 95929 Contributed by Gian-Carlo Rota, October 4, 1989

ABSTRACT A generalization is given of the notion of a alphabet L, written {alb}. The signature of a letter {aJb} is symmetric over a , which includes positive if both a and b are positively signed letters of L, it is variables of positive and negative signature ("supersymmetric negative if exactly one of a and b is negatively signed, and it variables"). It is shown that this structure is substantially is neutral if both a and b are negatively signed. Thus, the isomorphic to the exterior algebra of a vector space. A super- supersymmetric algebra Super[[L]'] is isomorphic to a symmetric extension of the second fundamental theorem of product of the symmetric algebra generated by the letters invariant theory is obtained as a corollary. The main technique {alb} for a and b negatively signed letters, the divided powers is a supersymmetric extension of the standard theorem. algebra generated by the divided powers of the letters {alb}, As a byproduct, it is shown that supersymmetric Hilbert space where a and b are positively signed, and the exterior algebra and supersymplectic space are in natural . generated by the letters {alb}, where the letters a and b are differently signed. The algebra ofsupersymmetric innerproducts, denoted by Section 1. Introduction Hilb[L], is defined as the quotient algebra of the supersym- metric algebra Super[[L]'] by the ideal generated by (i) all A Hilbert space ofdimension n over a k is a vector space elements {alb} + (-1)klaIbI{bja}, (ii) all elements {clc}, where c V, together with a nondegenerate symmetric bilinear k- is positively signed in L, and (iii) for both a and b positively valued form (vlv'), the inner product, defined for all vectors signed, then {alb}(k) - (-l)k{bla}(k). Denoting by (alb) the v and v' in V. Classical invariant theory gives a syntactic image of the letter {alb} in Hilb[L], we have in Hilb[L] the characterization of Hilbert space (at least when the field k is following commutation relations: (i) (alb) = -(bla) when not n] both a and b are negatively signed; (ii) (alb) = (bla) if both a infinite) by considering the commutative algebra Hilb[L-, and b are negatively signed; and (iii) (cdc) = 0 if c is positively generated by all ordered pairs (alb) of elements from an signed. The algebra of supersymmetric inner products is the alphabet L-, subject to the conditions (aib) = (bla) and supersymmetric algebra of formal inner products (alb) sub- det((ailbj): 1 . i, j . n + 1) = 0. The Second Fundamental ject to the above commutation relations and signature laws. Theorem of classical invariant theory asserts that the algebra If p is an element of Super[[L]'], we shall denote by Hilb[L-, n] can be faithfully realized as the algebra of inner Image(p) its canonical image in Hilb[L]. products in a suitable Hilbert space. Length, content, number of occurrences of letters of the Our objective is to generalize this result to the case of a alphabet L, as introduced in refs. 1 and 2, are obviously proper signed alphabet L (see refs. 1 and 2 for definitions). To extended to Super[[L]'] and to Hilb[L]. this end, we proceed in three steps. 1. The algebra of abstract inner products is generalized to Section 3. Hilbert Biproducts the algebra Hilb[L, n]. The generalization is the only possible one, since positively signed letters can be suitably polarized We shall use the notion of polarization in Super[L] as away to recover the classical case when all letters are introduced in section 4 of ref. 1. negatively signed. To every polarization D(b, a) of Super[L] one can 2. A standard basis theorem is established for the algebra associate a unique linear operator D(b, a) on Super[[L]'] by Hilb[L, n], using the notion of superstandard Young tableau. requiring that the operator D(b, a) be a (supersymmetric) As a byproduct of the supersymmetric generalization of the derivation of the algebra Hilb[L], having the same signature standard basis theorem, a duality is established between as D(b, a). More specifically, we have the following. supersymmetric Hilbert space and supersymplectic space. (i) IfD(b, a) is a positive polarization, set D(b, a)1 = 0; for 3. Finally, it is shown that the Second Fundamental The- all letters c, d in L set orem extends to the supersymmetric case. This is done by = + {c|D(b, a)d}. introducing the notion of model and by showing that, despite D(b, a){cld} {D(b, a)cJd} the appearance of variables of two signatures, there is a If both a and b are letters of the same sign, the divided natural way of mapping Hilb[L, n] to the algebra of inner powers D(k)(b, a) are defined as in section 5 of ref. 1. In products in an ordinary Hilbert space. particular, if a, b, and c are positive letters, one sets Section 2. Supersymmetric Inner Products D(k)(b, a){aicI}(r) = {blc}(k){alc}(r-k), when k s r, and 0 otherwise, whenever the letters a, b, and Let L be a proper signed alphabet and let Super[L] be the c are distinct. We further set of supersymmetric algebra generated by L, as defined in ref. 1. (for any signatures a, b, c, d) We define a third signed alphabet [L]' as follows. The letters D(2)(b, a){cld} = {D(b, a)clD(b, a)d}. of the alphabet [L]' are the ordered pairs a, b of letters of the If {cld} is a positive letter in [L]', set The publication costs of this article were defrayed in part by page charge D(k)(b, a){cld}(m) - payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact. IfD(b, a)clD(b, a)d}(')(D(b, a)fcid})(i)fcidJ(t), 653 Downloaded by guest on September 25, 2021 654 Mathematics: Rota and Stein Proc. Natl. Acad. Sci. USA 87 (1990) where the sum ranges over all nonnegative integers i, j, and (ii) If a and b are letters of the same sign, t satisfying i + j + t = m and 2i + j = k. For all monomials m and m' in Super[[L]'], set D(k)(b, a){wlw'} = X{D(b, a)(')wlD(b, a)(k-0)wI}, D(k)(b, a)(m m') = X(D(')(b, a)m)DO)(b, a)m', where the index i ranges between 0 and k. If D(b, a) is a negative polarization, then where the sum ranges over all i +j = k. When {alc} is a neutral letter and a #4 c, we have D(b, a){wlw'} = {D(b, a)wlw'} + (-)Iwl{wID(b, a)w'}. D(k)(b, a){ajc}' = (t!/(k!(t - k)!)){blc}k{alc}t-k (iii) {wlw'} = 0 unless Length(w) = Length(w'). and (iv) {1i1} = 1. (b) The composite ofthe preceding map with the canonical D(2)(b, a){ala} = {blb}, map ofSuper[[L]'] to Hilb[L] gives a w 0 w' -- when t 2 k and 0 otherwise. (In equation 5.6 of ref. 1, the (wiw') from Super[L] 0 Super[L] to Hilb[L], called the binomial coefficient is missing and should be inserted.) Hilbert biproduct, satisfying the following (not necessarily Example. Ifboth a and b are negatively signed letters, then independent) conditions. (i) If a and b are positively signed letters, then D(b, a)({alc}{ald}) = {blc}{ald} + {alc}{bld} and (a(k)Ib(k)) = (alb)(k). a D(2)(b, a)({alc}{ald}) = (ii) If is a positively signed letter, then (a(k)a(k)) = Ofor any {blc}{bld}. positive integer k. In the above two displayed formulas, all letters are assumed (iii) (WiW') = (-l)Letigth(w)(-l)lwllw'I(wIw). to be distinct. (iv) D(b, a)(wlw') = (D(b, a)wlw') + (-1)Leng'th(w)(-1)Iwjjw'I (ii) If D(b, a) is a negative polarization, set D(b, a)l = 0. (D(b, a)w'lw). For all letters c and d in L set (v) If a and b are letters of the same sign, then D(b, a){cjd} = {D(b, a)cld} + (-1)IcI{cID(b, a)d}; D(k)(b, a)(wlw') = Y:(D(')(b, a)wjD(k-i)(b, a)w') if-the letter {cld} is positively signed, set = (_ )Leh(w)(- )lwlwIIw'I(D(i)(b, a)w'ID(k-i)(b, a)w), D(b, a){cjd}(') = (D(b, a){cjd}){ccd}('1). where the index i ranges between 0 and k. For all monomials m and m' in Super[[Ll'], set (vi) (wlw') = 0 unless Length(w) = Length(w'). (vii) (111) = 1. D(b, a)(m m') = (D(b, a)m)m' + (-1)1m'mD(b, a)m'. When both w and w' are of Length n, we say that the Hilbert biproduct (wlw') is of length n. One shows that each polarization operator commutes with By abuse oflanguage, a product ofbiproducts, or ofHilbert the congruences defining the quotient algebra Hilb[L]. In biproducts, will be called a monomial. particular, one shows that D(b, a){cld} is congruent to {D(b, Example 1: Ifa1, a2,.. . , a)cld} f {-1)HIC^k{D(b, a)dlc}, as one easily verifies, giving the a,,, b1, b2,. .. , b,, are negatively identity in Hilb[L] signed letters, then D(b, a)(cld) = (D(b, a)cld) - (_j)tcIjdI(D(b, a)dlc). (aja2 ... anlbib2 ... bn) One therefore obtains polarization operators on Hilb[L], - D(al, u)D(a2, u) . . . D(an, u)D(bl, v)D(b2, v) which will be denoted by D(b, a). Polarization operators on Hilb[L] satisfy the same identities as those displayed above * . .D(bnq VO((n)lv(n)), for Super[[L]'], with parentheses replacing braces. We thus and the right side equals (-1)n(n-U1)/2 times the determinant of have the (aU) whose (commuting) entries are aij = (ailbj). In Image(Tp) = T(Image(p)) this case, the Hilbert biproduct reduces to the ordinary Hilbert space induced inner product. for any element p of Super[[L]'], where T is any product of Example 2: We have polarization operators. If w is a word in Div(L), one defines the wordpolarization (a(i)b(U)lU(k)v(n)) = Y(a(P)Iu(P))(a(r)lv(r))(b(q)lu(q))(b(s)lv(s)), operator T(w) in Super[L] as in ref. 1, as well as the corresponding operators T(w) in Super[[L]'] and T(w) in where the sum ranges over all nonnegative integers p, q, r, Hilb[L]. As in ref. 1, we write T(w, a) for the word polar- and s with p + q = k, r + s = n, p + r = i, and q + s = j. ization operator such that T(w, a)a(k) = w. This assumes that Example 3: Let j 5 n and i + j = k + n. The preceding Length(w) = k. expansion simplifies to For example, if u and v are positively signed letters, and if a, b, c, and d are negatively signed letters, we have (a(i)b(j)lb(k)c(n)) = (alb)(k)(alc)(n1)(blc)(J). [1] D(a, u)D(b, u)D(c, v)D(d, v)(ulv)(2) = (blc)(ald) - (alc)(bld). Since (alb)(k) = (-1)k(bla)(k), the right side equals The following basic result is easily established. (-1)k(bla)(k)(blc)(j)(alc)(ni); THEOREM 3.1. (a) There is a unique linear map w 0 w' -* {wlw'} from Super[L] 0 Super[L] to Super[[L]'], called the thus, by an application of Eq. 1 with b in place of a and a in biproduct, satisfying the following conditions. place of b, we obtain the important identity (i) If a and b are positively signed letters, then (i)b(j)lb(k)C(n))= (- 1)k(b(j+k)a(i-k)la(k)c(n)). [2] {a(k)lb(k)} = {alb}(k) Example 4: Using word polarization operators, we have the for any positive integer k. explicit formula for the biproduct Downloaded by guest on September 25, 2021 Mathematics: Rota and Stein Proc. Natl. Acad. Sci. USA 87 (1990) 655

(wlw') = T(w, a)T(w', b)(a(n)lb(n)). {w' w", w} = X sign(jw"Ijw(j)j){w', w(j)}{w", W(2)}. Let n be a positive integer. We define the supersymmetric An important special case is the following identity, which Hilbert algebra of n to be the quotient algebra of is used in the proof of Theorem 4.1 below: Hilb[L] obtained by dividing the algebra Hilb[L] by the ideal generated by all elements of the form Image(Tp), where T is {Pf(a(2r)), (Cld)(r)} = o any product of divided powers of polarization operators and for all positive integers r. p is any element of Super[[L]'] containing more than n The above definition ofthe pairing {w, w'} is consistent. We occurrences of some positively signed letter of L. now proceed to state the main result of this section. The supersymmetric Hilbert algebra of dimension n will be THEOREM 4.1. Let w be a word in Super[L] and let w', w" denoted by Hilb[L; n]. The image in Hilb[L; n] of the Hilbert be words in Super[L*]. Then biproduct (wlw') will again be denoted by (wlw'). By polarization of identity 2 we obtain the following. {Pflw), (w'lw")} = 0 THEOREM 3.2 (supersymmetric Kronecker identities). For any positive integer n we have in Hilb[L; n] unless Length(w) = 2 and Length(w') = Length(w") = 1 or else all words are of Length 0. X(w w,1)Jw,2) w") = (-1)IwIIwIl(wI w(i)Is(w(2)) w"). In the proof, one first establishes (case by case, consider- ing all possible signatures) the basic identity Recall that the antipode s is so defined that s(w) = (-)Length(')w. It can be shown that the Kronecker identities {Pf(abcd), (e*f*lg*h*)} = 0 give an alternative characterization of the supersymmetric Hilbert algebra of dimension n. and similar identities when some ofthe products are replaced by divided powers. The general case then follows by induc- Section 4. The Main Pairing tion. We have thus established a natural pairing between super- Recall the definition of the dual alphabet given in ref. 2: symplectic space and supersymmetric Hilbert space. In par- informally, we can say that the letters of L* are "the same" ticular, this pairing gives a duality between Symp[L'] and as the letters ofL but with opposite . Using the alphabet Hilb[L*-] and between Symp[L-] and Hilb[L*+]. It is thus [L] defined in ref. 1, we shall now establish a remarkable clear that such a pairing could not have been possible for pairing between the supersymplectic algebra and the Hilbert classical symplectic and Hilbert spaces, which correspond, in algebra. This pairing does not exist in the classical case; it is the present view, to taking negatively signed alphabets only. made possible by the use of supersymmetric variables. We shall apply the above result to the evaluation of Young More specifically, we shall define a from tableaux in both supersymplectic space and supersymmetric Symp[L] 0 Hilb[L*I to the Z of ordinary integers. Such Hilbert space. Let D* = (wl',W . Wk, wj) be a Young a linear form will be written {w, w'}, where w is an element diagram with an even number of rows on the alphabet L*. Set of Symp[L] and w' is an element of Hilb[L*]. By linearity, it suffices to define the linear form in the special case when both Hilb(D*) w and w' are the images in Symp[L] and Hilb[L*] of words in Div([L]) and Div([L*]'), respectively. We denote such images = (sign)(stand(w0)Jstand(wj)) ... (stand(wk)Istand(w')), by letters w and w'. where (sign) = sign(Elw,11Jwjj), and the sum ranges over all i The definition is carried out recursively in the following and j such that i < j. (deliberately redundant) steps. We call Hilb(D*) a Hilbert tableau. If D is a superstandard 1. Set {1, 1} = 1. Young diagram in L, the diagram D* in L* is defined as in ref. 2. Set {w, w'} = 0 if Length(w) 7 Length(w'). 2 and similarly for the dual diagram D-. 3. For a E L' and c*, d* E L*, set {Pf(a(2)), (c*ld*)} = 0, It can be verified that if D is a superstandard tableau all of unless c = d = a. whose columns are of even length, then Hilb(D) f 0 in 4. Set {Pf(ab), (c*lc*)} = 0, unless a = b = c. Hilb[L]. 5. Set {Pf(a(2)), (a*la*)} = 1 for a E L+. THEOREM 4.2. Let D and D' be Young diagrams in L. We 6. Set have {Pf(ab), (c*ld*)} {Pfaff(D), Hilb(D'*)} = 0 = (_ 1)1'jc 11(ajc*)(bjd*) + (- 1)jal bI(- 1)jC*IjaI(blc*)(ald*) unless the diagrams D and D' are a Gale-Ryser pair and, if 1 = (- 1)'jb1'1*(ajc*)(bjd*) ( ld)II'I(- )lblld*(ald*)(blc*). the diagram D is superstandard, then we have used the pairing introduced in section 3 of Here, (1) {PfafJ(D), Hilb(D-*)} = ±1. ref. 2. In other words, we set

{Pf(ab), (a*|b*)} = -1, Section 5. Standard Basis Theorem when a E L+ and b E L-, and Let D = (wl',W . Wk, wj) be a Young diagram (with an even number of rows) on the alphabet L. {Pf(ab), (a*lb*)} = 1 By analogy with the supersymplectic case treated in ref. 1 for all other signatures of the letters a and b. we have the following. 7. To define {w, w'} when w and w' are words of Length THEOREM 5.1 (standard basis theorem for supersymmetric greater than one, we use the method of supersymmetric Hilbert space). (a) There exist unique superstandard Young Laplace expansions, as in refs. 1-3. Specifically, set diagrams Di such that shape(D) - shape(D,) (in the domi- nance order of shapes) and unique nonzero integer coeffi- {w, w' w"} = I sign(jw(2)llw'l){w(l), W'}{W(2), w"} cients ci such that and Hilb(D) = I ciHilb(Di). Downloaded by guest on September 25, 2021 656 Mathematics: Rota and Stein Proc. Nati. Acad. Sci. USA 87 (1990) Each ofthe Di has the same content as D. Thus, the elements The tensor algebra Tens(Super[L]) over the supersymmet- Hilb(D), as D ranges over all superstandard tableaux, all of ric algebra Super[L] is generated as a module by the infinite whose columns are of even length, are an basis of tensor products Hilb[L]. (b) In Hilb[L; n] the elements Hilb(D), as D ranges over all W1 0 W2 0 W3 0 .... superstandard tableaux, all of whose columns are of even length and all of whose rows are of length at most equal to where w1, W2, W3, . . . are words in Super[L] and all but a n, form an integral basis. finite number of the wi equal 1. The product of two elements The proof that superstandard Hilbert tableaux span pro- ceeds by a straightening algorithm that uses the supersym- W1 i W2 0W3 ... and wj89 w, 8) W3 ... metric Kronecker identities together with the following (eas- is defined componentwise as ily proved) Laplace expansions for Hilbert biproducts: ±W1 Wj & W2 W2 W3 W3' 0..., (wlw' w") = L sign(Iw(2)IwI'I)(w(l)Iw')(w(2)Iw") where the sign is determined by the parity of the number of (w'w"lw) = X sign(lw"llw(l)I)(w' W(l))(W"IW(2)) transpositions of negative letters in going from the word w, leading to a straightening identity as in proposition 4 of W2 W3 ... W1 ... to the word w1 w' W2 W2 W3 W3 .... ref. 2. The is similarly defined componentwise. The proof that the superstandard Hilbert tableaux are The notation for Tens(Super[L]) is cumber- independent uses Theorem 4.2, which establishes a triangular some, and it is a relief to realize that it can be replaced by a relationship between superstandard supersymplectic tab- more pliable notation. leaux and superstandard Hilbert tableaux. For the case of PROPOSITION 6.2. The time-ordered supersymmetric alge- Hilb[L; n], one may adapt the method of polarization intro- bra Super[LIP] is naturally isomorphic to the tensor algebra duced in ref. 1. Tens(Super[L]). We observe that the preceding argument gives yet another The natural is the one that maps the mono- proof that superstandard symplectic tableaux span (which is mial m in time-ordered form a part of theorem 4 of ref. 1), which is simpler than the one given in ref. 1, though not explicitly algorithmic. Mt=(the(emn2t2 3 3 Section 6. Time-Ordering to the element

A time-ordered supersymmetric algebra is a supersymmetric W1 0 W2 0 W3 0 algebra Super[LIP], where P = P+ is a positively signed alphabet, ordered by the order type of the positive integers. of Tens(Super[L]). PROPOSITION 6.1. Every element m' ofSuper[LIP], homo- In view of Proposition 6.2, we identify the tensor algebra geneous in each of the letters of L occurring in m', can be Tens(Super[L]) with the time-ordered algebra Super[LIP], uniquely written as an integral of mono- and sometimes refer to the letters in the alphabet P as places. mials An argument similar to that which has been used in the proof of Proposition 6.1 proves the following. m = (wIp(i))(w'Iq6))(wIIr(k)) .. .wIS), PROPOSITION 6.3. Let L = L+; in other words, let all letters of L be positively signed. Then an integral basis of every where p, q, ... . r, s are letters in P such that p < q < ... < submodule ofSuper[LIP] spanned by all words with a given r < s and where w, w',. .. . w", w"' are any words in Super[L]. content is given by all monomials of the form In each such monomial m every letter occurs with the same degree as it occurs in m'. mir (a01~w)(b0)1W') . .. (C(k)1w'), Sketch of proof. Any term of Super[LIP] of the form (ulv v'), where u is a word in Super[L] and v v' are words in where w, w', . . , w" are words in Super[P], and where a, b, Super[P] can be expanded, by a Laplace expansion, into a . . ., c are letters ofthe alphabet L such that a < b < ... < linear combination of products X(u(l)Iv)(u(2)Iv'). Expanding c in the ordered set L. successively in this way, one eventually obtains linear com- As a consequence of Proposition 6.3 we have the follow- binations of monomials of the form (wlp(i))(w'Iq(i))(w11jr(k) ing. ... (w"'.Is"'), where the letters p, q, ... , r, s are not PROPOSITION 6.4. Let B be a basis ofSuper[L-IP]. Then a necessarily distinct. Thus, monomials of this form are a set basis of Super[LIP] is given by the set of monomials bm, as of generators. b ranges over the basis B and m ranges over all monomials But if, say, p = q, then repeated application of the Laplace specified in Proposition 6.3. identity (wjp('))(w'Jq(')) = (w w'lp(i+j)) simplifies the expres- sion, until one obtains a monomial m as in the statement of Section 7. Models

Proposition 6.1, with distinct letters p, q, . . . r, s. Finally, one permutes the monomials (wjp(i)), (wtlq(U), (wI|r(k)), . . We consider the following universal problem. Given alpha- (w"'Is(')) (thereby introducing some signs) until the letters p, q, bets L and L' = L' (in other words, L' is negatively signed) r, . ..,s are in increasing order. and given a function 4 from L to Super[L'] such that 4(a) is A monomial such as m above can be rewritten in time- a homogeneous element of Super[L'] for every positively ordered form as an infinite product: signed letter a in L, and 4(a) is a letter of L' for every negatively signed letter a, find an extension of the map 4 to (W=( Uyltl)(W IplI 2))(W, II r 3)). a map 4 that is nonzero on a "maximal" submodule of Super[LIP], with values in Super[L'IP]. The solution to this where P1i P2, p3,... are the first, second, third, ... letters universal problem is obtained as follows. of the alphabet P; almost all words wk equal 1; and almost all First, if4k(a) is a homogeneous element ofdegree k, say that exponents i. equal 0 (recall that p°) = 1). the letter a has arity equal to k. Downloaded by guest on September 25, 2021 Mathematics: Rota and Stein Proc. Natl. Acad. Sci. USA 87 (1990) 657

A letter of arity k occurring in a monomial m in Super[LIP] Hilb((D): Hilb(m) -* Hilb((D(m)) is said to occur in m with proper arity if it occurs in m with multiplicity equal to its arity. well-defines a linear map from Hilb[L; n] to Hilb[L'; n]. A monomial m in Super[LIP] is said to have proper arity if THEOREM 8.1 (second fundamental theorem for supersym- every positively signed letter that occurs in m occurs in m metric Hilbert space). Let the (negatively signed) alphabet L' with proper arity. No restriction is placed on negatively have n letters, and suppose that the element p ofHilb[L; n] signed letters. has the property that The umbral space Umb[O] is the submodule spanned by all Hilb((D)(p) = 0 monomials having proper arity. A linear operator (F from Umb[O] to Super[L'IP], extending for every model (F of Super[L'IP]. Then p = 0. the map 4, is defined as follows. We shall give some applications of Theorem 8.1. Recall that 1. If +(a) = t, and a is a positively signed letter of arity k a Hilbert space of dimension n over a field k is a vector space (i.e., the element t is homogeneous of degree k), set V, together with a nondegenerate symmetric bilinear form (((a(k)Iw)) = for every w in Super[P]. If a is a negatively (vlv') = (v'lv) (sometimes called the inner product) defined for ((jw) all vectors v, v' in V and taking values in the field k. signed letter in L, set (F((ajw)) = (O(a)lw) for any w in The exterior algebra Ext(V) over the Hilbert space V is Super[P]. isomorphic to the supersymmetric algebra Super[{vj, v2, 2. If v,}], where the letters v1, v2,. . . , v,, are assumed to be negative. The set {vj, v2, . . , vj} can be mapped to any basis m"= (a(')Iw)(b(0)Iw') . .. (c(k)w") of the Hilbert space V. We shall write Super(VIP] in place of is a monomial in Super[LIP] with a < b < . < c, and with Super[{vj, v2,.. . v, }IP]. Clearly Super(VIP] is independent a, ..., c positively signed letters, set of the choice of a basis. b, Example 1: Let L be a negatively signed set of sufficiently = ... high cardinality. Let 4 map the elements of L to vectors in V, VW(m") (O(a)jw)(O(b)Iw') (O(c)jw"). in any way. Theorem 8.1 asserts that an identity among 3. If m' is a monomial whose L-content consists only of "abstract" inner products in Hilb[L, n] holds for "generic" negative letters, and if m is any monomial, set (F(mm') = vectors if and only if it holds in an actual Hilbert space. (F(m)4(m'). Example 2: Let p be a polynomial in Hilb[L; n] containing 4. Extend by linearity to all of Umb(O). Since the subspace positive letters a, b, .. . , c of arities i, j, . . ., k, as well as is a direct summand negatively signed letters. Theorem 8.1 asserts that p = 0 in Umb(O) of Super[LIP], we may assume Hilb[L; n] ifand only ifHilb((D)(p) = 0 for all models (D. Now, that (F is defined on all of Super[LIP] and (I(p) = 0 for p in a model assigns to each of the letters a, b, . , c skew- the complement of Umb(O). symmetric tensors t, t', . .. , t" of steps ij,. . , k. Thus, the The triple consisting of the alphabet L, the umbral space p = 0 is an identity that holds for arbitrary choice of tensors Umb(O), and the map (F is said to be a model of the t, t', . . . , t". The correspondence between positively signed supersymmetric algebra Super[L'jP]. In 19th century usage, letters and skew-symmetric tensors that is given by a model one improperly says that the alphabet L is a symbolic (or allows the use of a "generic letter" such as a(k) to denote a umbral) representation of the supersymmetric algebra skew-symmetric tensor, without writing such a tensor as the Super[L'IP]. Note that, in the case we are considering, the sum ofdecomposable tensors, and to derive identities for the supersymmetric algebra Super[L'IP] is an exterior algebra. algebra of skew-symmetric tensors by manipulating a com- Example: If m" = (a(k)Ipy)p(i)), where i + i = k, and (a) = mutative algebra, namely, the divided powers algebra. w, say, then Addendum (F(m") = w())(W(2)1P() = I' W(1) 09 W(2), The definition of binomial coefficients in the statement of where the sum Y' ranges over all pairs w(l) and W(2) of lengths theorem 2 in ref. 1 should be modified as follows. i and j, respectively. Set Section 8. Models in Hilbert Space (O 1 p ) n)=°( - n)=° We denote by Hilb(w) the canonical image of an element of Super[LIP] in Hilb[L; n]. If m is a time-ordered monomial where n is a positive integer and p is a nonnegative integer. The work ofG.-C.R. is supported by National Science Foundation m,= (@'1Ipit))(W42'p,~i'))Cv@'2i3)) Grant MCS 8104855. The work ofJ.A.S. is supported by agrant from the Sloan Foundation. we denote by Hilb(m) the element 1. Rota, G.-C. & Stein, J. A. (1989) Proc. Nall. Acad. Sci. USA 86, = Hilb(m) (Wi1w2)(w31w4) .. 2521-2524. 2. Rota, G.-C. & Stein, J. A. (1986) Proc. Nail. Acad. Sci. USA 83, of Hilb[L; n]. 844-847. LEMMA 8.1. Let the (negatively signed) alphabet L' have 3. Grosshans, F., Rota, G.-C. & Stein, J. A. (1987) Invariant n letters, and let (F be a model of Super[L'IP]. Then Theory and Superalgebras (Am. Math. Soc., Providence, RI). Downloaded by guest on September 25, 2021