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Proc. Nati. Acad. Sci. USA Vol. 87, pp. 653-657, January 1990 Mathematics Supersymmetric Hilbert space (supersymmetric algebra/algebras with straightening laws/Young tableaux/invariant theory, skew-symmetric tensors) 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 bilinear form over a vector space, 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 tensor 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 basis 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 duality. 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 field 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 operator 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 linear map 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].
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