It is shown that the Hirota derivative can be used to construct the plethysm for tensor products of representations of {sl}_2(k)

Athorne, C.

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Chris Athorne

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Hirota derivatives and representation theory

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       Athorne, C. (2001) Hirota derivatives and representation theory. Glasgow Mathematical Journal, 43 (A). pp. 1-8. ISSN 0017-0895  http://eprints.gla.ac.uk/13352/  Deposited on: 19 April 2010   Enlighten – Research publications by members of the University of Glasgow http://eprints.gla.ac.uk HIROTA DERIVATIVES AND REPRESENTATION THEORYCHRIS ATHORNEDepartment of Mathematics, University of Glasgow, Glasgow G12 8QW, UKe-mail: ca@maths.gla.ac.uk(Received 14 March, 2000)Abstract. It is shown that the Hirota derivative can be used to construct theplethysm for tensor products of representations of sl2ðkÞ:1991 Mathematics Subject Classiﬁcation. 35Q51, 17B80.1. Introduction. The Hirota ‘derivative’ [5] is a process of considerable power inthe theory of solitons and integrable systems generally. Its use allows the repre-sentation of many integrable equations in bilinear form and thereby simpliﬁes theanalytic process of the inverse scattering transform [6] to an algebraic procedure fora large class of solutions. It retains nevertheless a certain mystique. The aim of thispaper is to unveil the Hirota derivative, just a fraction, by exploring its connectionwith representation theory [1,9].We start by formulating the Hirota derivative as a map of tensor products ofsl2ðkÞ-modules, where k is any ﬁeld of zero characteristic, and showing that it acts asa partial intertwining operator for the action of this algebra. The main thrust of thepaper is to show that the entire decomposition of the tensor product into irreduciblerepresentations can be accomplished by applying the Hirota map to a speciﬁc lowestweight vector. This is achieved by creating a new sl2ðkÞ-module, a homogeneous,graded algebra, to which standard results can be applied. On the way certain inter-esting Hilbert-Poincare´ series are presented.2. The Hirota derivative. Let k be a ﬁeld of characteristic 0 and denote by e, fand h the k-basis of the algebra sl2ðkÞ satisfying the Lie product identities:½e; f  ¼ h; ½h; e ¼ 2e; ½h; f  ¼ 2f:By the sl2ðkÞ-module An we mean the standard nþ 1 dimensional representation ofsl2ðkÞ with basis fa0; a1; . . . ; ang with respect to which e, f and h may be taken to bediﬀerential operators:e ¼ na1 @@a0þ ðn 1Þa2 @@a1þ . . .þ an @@an1;f ¼ a0 @@a1þ 2a1 @@a2þ . . .þ nan1 @@an;h ¼ nan @@anþ ðn 2Þan1 @@an1þ . . . na0 @@a0:Glasgow Math. J. 43A (2001) 1–8. # Glasgow Mathematical Journal Trust 2001. Printed in the United KingdomLet Agn be the g-fold k-tensor product of An. We shall write elements of thistensor product as k-linear combinations of lists of g integers (between 0 and n) underthe correspondence:ak1  ak2  . . . akg $ ðk1; k2; . . . ; kgÞ:The operators e, f and h are taken to extend to Agn in the usual fashion (Leibnitzrule). Agn decomposes as a direct sum of eigenspaces for h. The associated eigen-values are the weights.Definition 1. The isobaric weight of ðk1; . . . ; kgÞ 2 Agn is p ¼ k1 þ k2 þ . . .þ kgÞ.The weight of this element is 2p ng.The eﬀect of e ( f ) is to increase (decrease) the isobaric weight of an isobaricelement by one and hence the weight by two.We now introduce the key player.Definition 2. The Hirota map Dij : Agn !Agnþ1 is deﬁned by k-linear extensionof its action on a basis, an action given by:Dij ð. . . ; ki; . . . ; kj . . .Þ ¼ ð. . . ; ki þ 1; . . . ; kj . . .Þ  ð. . . ; ki; . . . ; kj þ 1 . . .Þ ð1ÞNote that there are only g 1 linearly independent Dij and that the map isantisymmetric in its two indices. Dij increases the isobaric weight of an isobaric ele-ment by one but the weight by 2 g.The original form of the Hirota map (‘derivative’) [5] appears in the theory ofsoliton equations where its remarkable properties make it of prime importance tothe explicit description of soliton solutions [8]. The deﬁnition given above is actuallyassociated with a generalized Hirota derivative [3] though the current form of itspresentation is due to [1]. The link to algebraic invariants, and hence representationtheory, is ampliﬁed in [1] and remarked upon in [9].From our point of view the most important property of the Hirota map is that itis a partial intertwining operator for the sl2ðkÞ-modules Agn and Agnþ1. This may beexpressed precisely as follows.Lemma 3. The maps Dij : Agn !Agnþ1 satisfy the following commutation rules withf and h:½Dij; f  ¼ 0; ð2Þ½Dij; h ¼ ðg 2ÞDij: ð3ÞNote that in the lemma f and h act on both Agn and Agnþ1. There appears to be nosimple expression for the commutator of Dij with e.The chief consequence of the partial intertwining relations is that elements of Agnwhich are killed by f, that is lowest weight vectors, are mapped by the Dij into lowestweight vectors in Agnþ1. Since we know that Agn decomposes into irreducible repre-sentations of sl2ðkÞ the Hirota map will generate associated lowest weight vectorsfrom the most basic one ð0; . . . ; 0Þ 2 Agn. The question which we want to address is2 CHRIS ATHORNEwhether it generates the lowest weight vectors of all the irreducibles in Agn. We shallshow that the answer is yes by a counting argument. Before we do so, we give acouple of simple examples.3. Examples of plethysm.A32The basic element ð0; 0; 0Þ has weight -6 and is the lowest weight vector for a 7dimensional irreducible representation. Now consider ð0; 0; 0Þ 2 A31. There are twolinearly independent highest weight vectors in A32, D12ð0; 0; 0Þ ¼ ð1; 0; 0Þ  ð0; 1; 0Þand D13ð0; 0; 0Þ ¼ ð1; 0; 0Þ  ð0; 0; 1Þ each of weight -4 and giving 5 dimensionalirreducible representations. For ð0; 0; 0Þ 2 A30 each of D212ð0; 0; 0Þ ¼ ð2; 0; 0Þ2ð1; 1; 0Þ þ ð0; 2; 0Þ, D213ð0; 0; 0Þ¼ð2; 0; 0Þ  2ð1; 0; 1Þ þ ð0; 0; 2Þ and D12D13ð0; 0; 0Þ¼ð2; 0; 0Þ  ð1; 1; 0Þ  ð1; 0; 1Þ þ ð0; 1; 1Þ generate the 3 dimensional representationsand ﬁnally D12D23D31ð0; 0; 0Þ ¼ ð2; 1; 0Þ  ð1; 0; 2Þ þ ð2; 0; 1Þ  ð0; 2; 1Þ þ ð1; 2; 0Þþð0; 1; 2Þ belongs to A32, as a subspace of A33, has weight 0 and is a one dimensionalrepresentation (invariant). These are all the lowest weight vectors which can begenerated using the Hirota maps and it is easy to see that they give the completedecomposition by summing over dimensions with multiplicities:3 3 3 ¼ 7 52  33  1:A42Brieﬂy, we have a 9 dimensional representation with lowest weight vectorð0; 0; 0; 0Þ 2 A42; three 7 dimensional representations coming from D12, D13 and D14acting on ð0; 0; 0; 0Þ 2 A41; six 5 dimensional representations from D212, D213, D214,D12D13, D12D14 and D14D13; six 3 dimensional representations coming from D212D34,D12D234, D13D224, D213D24, D14D232 and D214D32; and three 1 dimensional coming fromD212D234, D213D224 and D214D232. This is precisely the decomposition,3 3 3 3 ¼ 9 73  56  36  13:One may do many such examples by hand, reinforcing the conjecture thatthe Hirota map and the action of e construct all the irreducibles in the tensorproduct.4. Counting lowest weight vectors. If our conjecture is correct the lowest weightvectors in Agn will come from polynomials in the Dij which, when applied toð0; 0; . . . ; 0Þ, never yield an integer greater than n. This is the same as seeking poly-nomials over k in the diﬀerences xi  xj of g variables x1; . . . ; xg which involve nopower of any xi greater than n.Let R be the polynomial ring k½x1; . . . ; xg, I the ideal of R generated by the setfxnþ11 ; . . . ; xnþ1g g and J the ideal of R generated by the element x1 þ x2 þ . . .þ xg. Wehave the short exact sequence:I,!R! R=I: ð4ÞLet  be the map,HIROTA DERIVATIVES AND REPRESENTATION THEORY 3 : R! R; xi 7!xi  1gðx1 þ x2 þ . . .þ xgÞwhich has kernel J and let  be the projection : R! R=J:This is precisely the polynomial ring in the diﬀerences of the xi. Now apply  to (4).It is easy to see that we obtain the short exact sequence,II \ J ,!RJ! ðR=J ÞI=ðI \ J Þand, by the isomorphism theorems,ðR=J ÞI=ðI \ J Þ ﬃðR=J ÞðIþ J Þ=J ﬃRIþ Jwhich is the homogeneous (ﬁnite dimensional) k-algebraKng ¼Rðxnþ11 ; . . . xnþ1g ; x1 þ x2 þ ::þ xgÞR: ð5ÞThis homogeneous algebra is graded by isobaric weightKng ¼Mngp¼0Kn;pgand the k-dimension of each component is the number of lowest weight vectorsconstructed by use of Hirota maps. (Note that for p larger than some value between0 and ng these dimensions are zero.) Each comes with an irreducible representationof dimension ng 2pþ 1 and so we have to show thatXngp¼0ðng 2pþ 1Þ dimKn;pg ¼ ðnþ 1Þg ð6Þwhich is the dimension of Agn. This is the same as showing thatðngþ 1Þkngð1Þ  2k 0ngð1Þ ¼ ðnþ 1Þgwhere kngðsÞ is the Hilbert-Poincare´ series of Kng,kngðsÞ ¼Xngp¼0s p dimKn;pgand prime denotes the derivative with respect to s.The assertion is relatively easily veriﬁed in particular instances using algorithmsbased on Gro¨bner bases implemented in software packages like MAPLE [7]. How-ever, the proof is not so easily accomplished for general n and g.4 CHRIS ATHORNE5. The Hilbert-Poincare´ Series. This section constitutes the meat of the presentpaper and is devoted to obtaining a closed form expression for the Hilbert-Poincare´series, kngðsÞ, of Kng. The algebra is constructed as the kernel of a diﬀerential operatoracting on a simpler graded algebra. Let hngðsÞ denote the Poincare´-Hilbert series ofthe homogeneous algebra R=I and let  denote the diﬀerential operator@x1 þ @x2 þ . . . @xg .Lemma 4.  maps R=I to itself with kernel R=ðIþ JÞ.Proof. This is a straightforward consequence of the theory of partial diﬀerentialequations with constant coeﬃcients.Lemma 5. The isobaric weight gives a grading of R=IR=I ¼Mngp¼0Mn;pg(every component of which is nontrivial) and  maps Mn;pg to Mn;p1g :Proof. The grading is obvious and since the element a ¼ ðx1 . . . xgÞn provides abasis for Mn;ngg and nga ¼ ðngÞ! 6¼ 0 we see that all the components are non trivial.Lemma 6. The Hilbert-Poincare´ series of R=I ishngðsÞ ¼ ð1þ sþ s2 þ . . .þ snÞgProof. By induction on g:Mn;pg ¼Mn;pg1  xgMn;p1g1  x2gMn;p2g2  . . . xngMn;png1and hencehngðsÞ ¼ hng1ðsÞ þ shng1 þ s2hng1 þ . . .þ snhng1and it is straightforward thathn2 ¼ 1þ sþ s2 þ . . .þ sn:The idea now is to show that R=I actually carries a representation of sl2ðkÞ inwhich Kng appears as the span of lowest weight vectors. We achieve this by means ofthe following constructions very loosely analogous to the Hodge star and cobound-ary operations in the theory of de Rham complexes.Definition 7. Let a be a monomial element of R=I. We deﬁne a to be themonomial with the property a  a ¼ ðx1x2 . . . xgÞn. Now extend  by k-linearity tothe general element of R=I.  thus maps Mn;pg to Mn;ngpg .HIROTA DERIVATIVES AND REPRESENTATION THEORY 5It is easily seen, by the symmetry of the Hilbert-Poincare´ series,snghngð1=sÞ ¼ hngðsÞthat  is an isomorphism of k-vector spaces.Definition 8. With the map  : Mn;pg !Mn;p1g we associate a ‘‘dual’’ map : Mn;pg !Mn;pþ1g deﬁned by ¼  ð7ÞLemma 9. The operators ,  and H ¼  make R=I into an sl2ðkÞ-module.Proof. By considering the action of the operators on a monomial element it caneasily be seen that: ¼ nðx1 þ . . .þ xgÞ  x21@1  . . . x2g@gH ¼ ng 2ðx1@1 þ . . .þ xg@gÞ½H; ¼ 2½H;  ¼ 2Theorem 10. R=I decomposes into a direct sum of irreducible sl2ðkÞ-modules iso-morphic to Ad of dimensions d ¼ ngþ 1; ng 1; ng 3; . . . with multiplicities	d ¼ dimMn;ðdþng1Þ=2g  dimMn;ðdþngþ1Þ=2g :Proof. This follows from standard representation theory [2]. The Mn;pg are theeigenspaces of H with weights ng 2p. Each lowest weight vector, v with ðvÞ ¼ 0, ofweight ng 2p belongs to an irreducible representation isomorphic to Ad of dimen-sion d ¼ 2p ngþ 1: The highest dimensional irreducible is associated with thelowest weight vector ðx1 . . . xgÞn 2Mn;ngg and has dimension ngþ 1. Factoring outthis irreducible we are left with dimMn;ng1g  dimMn;ngg lowest weight vectors inMn;ng1g each generating irreducibles of dimension ng 1. Factoring out these andcontinuing we arrive at the result.A consequence of this is that, because of the symmetry between highest andlowest weight vectors, we can easily construct the kernel of  as the span of thehighest weight vectors and this gives the result:Theorem 11. The Hilbert-Poincare´ series of R=ðIþ J Þ iskngðsÞ ¼ ½ð1 sÞhgnðsÞþ ð8Þwhere ½þ denotes the truncation to the terms with positive coeﬃcients.6 CHRIS ATHORNEFinally, we see that the formula (6) for the total dimensions of all the irreduciblerepresentations generated by the Hirota maps is just same as the formula for thetotal dimension of all the irreducibles in R=I, that is the dimension of R=I. Hence:Theorem 12.ðngþ 1Þkngð1Þ  2k 0ngð1Þ ¼ hngð1Þ ¼ ðnþ 1Þg: ð9ÞThe approach above immediately generalizes to the g-fold tensor product ofsl2ðkÞ-modules of arbitrary dimensions:Agn1;...;ng ¼ An1  An2  . . . Ang :We therefore consider the homogeneous algebraR=ðxn1þ11 ; . . . ; xngþ1g ÞR;whose Hilbert-Poincare´ polynomial ishn1;n2;...;ngðsÞ ¼Ygi¼1ð1þ sþ . . .þ s niÞand which carries a sl2ðkÞ-representation given by the diﬀerential operators ¼ @1 þ . . . @g and  ¼ , where  is deﬁned by its k-linear extension from itsaction on monomials a, namely,aðaÞ ¼ xn11 xn22 . . . xngg :Explicitly,  is given by ¼ n1x1 þ . . .þ ngxg  x21@1  . . . x2g@gandH ¼ ½;  ¼ n1 þ . . .þ ng  2ðx1@1 . . . xg@gÞ:,  and H are easily seen to satisfy the commutation relations of sl2ðkÞ. Again theHilbert-Poincare´ polynomial is ½ð1 sÞQgi¼1ð1þ sþ . . .þ sniÞþ and the entiredecomposition arises from Hirota maps, because the dimensions satisfy the formulaðn1 þ . . .þ ng þ 1Þkð1Þ  2k0ð1Þ ¼Ygi¼1ðni þ 1Þ:6. Conclusion. We have presented a derivation of Hilbert-Poincare´ series forcertain homogeneous algebras generated by the generalized Hirota map and shownthereby that such maps generate the entire decomposition of tensor products ofstandard representations into irreducibles. This can be regarded as an extension ofthe constructive methods of classical invariant theory [9].HIROTA DERIVATIVES AND REPRESENTATION THEORY 7There are two obvious directions for further work.Firstly, given the importance of the Hirota derivative in integrable systems the-ory, one may ask whether the above results indicate an ‘explanation’ for this enig-matic phenomenon. There are, for instance, connections with the Moyal bracket ofquantum mechanics [10].Secondly, to what extent can this simple and attractive algebraic structure begeneralised to other Lie algebras or to diﬀerential operators other than ; forexample, operators of second order?Finally, I should like to thank Ken Brown for one or two critical ears.REFERENCES1. C. Athorne, Algebraic invariants and generalized Hirota derivatives, Phys. Lett. A256 (1999), 20–24.2. W. Fulton and J. Harris, Representation theory (Springer-Verlag, 1991).3. B. Grammaticos, A. Ramani and J. Hietarinta, Multilinear operators: the naturalextension of Hirota’s bilinear formalism, Phys. Lett. A 190 (1994), 65–70.4. D. Hilbert, Lectures on algebraic invariants (Cambridge University Press, 1993).5. Ryogo Hirota, Bilinearization of soliton equations, J. Phys. Soc. Japan 51 (1982),323–331.6. S. V. Manakov, S. Novikov, L. P. Pitaevskii and V. E. Zakharov, Theory of solitons(Plenum, 1984).7. Maple V Release 5.00 (Waterloo Maple Inc., 1997).8. J. J. C. Nimmo, Hirota’s method, in Soliton theory: a survey of results, NonlinearScience: Theory and Applications (Manchester University Press, Manchester, 1990), 75–96.9. P. J. Olver, Classical invariant theory (Cambridge University Press, 1999).10. I. A. B. Strachan, A geometry for multidimensional integrable systems, J. Geom.Phys. 21 (1997), 255–278.8 CHRIS ATHORNE

A geometry for multidimensional integrable systems,

Algebraic invariantsand generalized Hirota derivatives ,

Bilinearization of soliton equations,

Classical invariant theory

Hirota’smethod, in Soliton theory: a survey of results, Nonlinear Science: Theory and Applications

Lectures on algebraic invariants

Multilinear operators: the natural extension of Hirota’s bilinear formalism,

Release 5.00

Representation theory (Springer-Verlag,

Theory of solitons (Plenum,

Enlighten: Publications

 
 
 
 
 
 
 
Athorne, C. (2001) Hirota derivatives and representation theory. 
Glasgow Mathematical Journal, 43 (A). pp. 1-8. ISSN 0017-0895 
 
http://eprints.gla.ac.uk/13352/ 
 
Deposited on: 19 April 2010 
 
 
Enlighten – Research publications by members of the University of Glasgow 
http://eprints.gla.ac.uk 
HIROTA DERIVATIVES AND REPRESENTATION THEORY
CHRIS ATHORNE
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK
e-mail: ca@maths.gla.ac.uk
(Received 14 March, 2000)
Abstract. It is shown that the Hirota derivative can be used to construct the
plethysm for tensor products of representations of sl2ðkÞ:
1991 Mathematics Subject Classification. 35Q51, 17B80.
1. Introduction. The Hirota ‘derivative’ [5] is a process of considerable power in
the theory of solitons and integrable systems generally. Its use allows the repre-
sentation of many integrable equations in bilinear form and thereby simplifies the
analytic process of the inverse scattering transform [6] to an algebraic procedure for
a large class of solutions. It retains nevertheless a certain mystique. The aim of this
paper is to unveil the Hirota derivative, just a fraction, by exploring its connection
with representation theory [1,9].
We start by formulating the Hirota derivative as a map of tensor products of
sl2ðkÞ-modules, where k is any field of zero characteristic, and showing that it acts as
a partial intertwining operator for the action of this algebra. The main thrust of the
paper is to show that the entire decomposition of the tensor product into irreducible
representations can be accomplished by applying the Hirota map to a specific lowest
weight vector. This is achieved by creating a new sl2ðkÞ-module, a homogeneous,
graded algebra, to which standard results can be applied. On the way certain inter-
esting Hilbert-Poincare´ series are presented.
2. The Hirota derivative. Let k be a field of characteristic 0 and denote by e, f
and h the k-basis of the algebra sl2ðkÞ satisfying the Lie product identities:
½e; f  ¼ h; ½h; e ¼ 2e; ½h; f  ¼ 2f:
By the sl2ðkÞ-module An we mean the standard nþ 1 dimensional representation of
sl2ðkÞ with basis fa0; a1; . . . ; ang with respect to which e, f and h may be taken to be
differential operators:
e ¼ na1 @
@a0
þ ðn 1Þa2 @
@a1
þ . . .þ an @
@an1
;
f ¼ a0 @
@a1
þ 2a1 @
@a2
þ . . .þ nan1 @
@an
;
h ¼ nan @
@an
þ ðn 2Þan1 @
@an1
þ . . . na0 @
@a0
:
Glasgow Math. J. 43A (2001) 1–8. # Glasgow Mathematical Journal Trust 2001. Printed in the United Kingdom
Let Agn be the g-fold k-tensor product of An. We shall write elements of this
tensor product as k-linear combinations of lists of g integers (between 0 and n) under
the correspondence:
ak1  ak2  . . . akg $ ðk1; k2; . . . ; kgÞ:
The operators e, f and h are taken to extend to Agn in the usual fashion (Leibnitz
rule). Agn decomposes as a direct sum of eigenspaces for h. The associated eigen-
values are the weights.
Definition 1. The isobaric weight of ðk1; . . . ; kgÞ 2 Agn is p ¼ k1 þ k2 þ . . .þ kgÞ.
The weight of this element is 2p ng.
The effect of e ( f ) is to increase (decrease) the isobaric weight of an isobaric
element by one and hence the weight by two.
We now introduce the key player.
Definition 2. The Hirota map Dij : Agn !Agnþ1 is defined by k-linear extension
of its action on a basis, an action given by:
Dij ð. . . ; ki; . . . ; kj . . .Þ ¼ ð. . . ; ki þ 1; . . . ; kj . . .Þ  ð. . . ; ki; . . . ; kj þ 1 . . .Þ ð1Þ
Note that there are only g 1 linearly independent Dij and that the map is
antisymmetric in its two indices. Dij increases the isobaric weight of an isobaric ele-
ment by one but the weight by 2 g.
The original form of the Hirota map (‘derivative’) [5] appears in the theory of
soliton equations where its remarkable properties make it of prime importance to
the explicit description of soliton solutions [8]. The definition given above is actually
associated with a generalized Hirota derivative [3] though the current form of its
presentation is due to [1]. The link to algebraic invariants, and hence representation
theory, is amplified in [1] and remarked upon in [9].
From our point of view the most important property of the Hirota map is that it
is a partial intertwining operator for the sl2ðkÞ-modules Agn and Agnþ1. This may be
expressed precisely as follows.
Lemma 3. The maps Dij : Agn !Agnþ1 satisfy the following commutation rules with
f and h:
½Dij; f  ¼ 0; ð2Þ
½Dij; h ¼ ðg 2ÞDij: ð3Þ
Note that in the lemma f and h act on both Agn and Agnþ1. There appears to be no
simple expression for the commutator of Dij with e.
The chief consequence of the partial intertwining relations is that elements of Agn
which are killed by f, that is lowest weight vectors, are mapped by the Dij into lowest
weight vectors in Agnþ1. Since we know that Agn decomposes into irreducible repre-
sentations of sl2ðkÞ the Hirota map will generate associated lowest weight vectors
from the most basic one ð0; . . . ; 0Þ 2 Agn. The question which we want to address is
2 CHRIS ATHORNE
whether it generates the lowest weight vectors of all the irreducibles in Agn. We shall
show that the answer is yes by a counting argument. Before we do so, we give a
couple of simple examples.
3. Examples of plethysm.
A32
The basic element ð0; 0; 0Þ has weight -6 and is the lowest weight vector for a 7
dimensional irreducible representation. Now consider ð0; 0; 0Þ 2 A31. There are two
linearly independent highest weight vectors in A32, D12ð0; 0; 0Þ ¼ ð1; 0; 0Þ  ð0; 1; 0Þ
and D13ð0; 0; 0Þ ¼ ð1; 0; 0Þ  ð0; 0; 1Þ each of weight -4 and giving 5 dimensional
irreducible representations. For ð0; 0; 0Þ 2 A30 each of D212ð0; 0; 0Þ ¼ ð2; 0; 0Þ
2ð1; 1; 0Þ þ ð0; 2; 0Þ, D213ð0; 0; 0Þ¼ð2; 0; 0Þ  2ð1; 0; 1Þ þ ð0; 0; 2Þ and D12D13ð0; 0; 0Þ¼
ð2; 0; 0Þ  ð1; 1; 0Þ  ð1; 0; 1Þ þ ð0; 1; 1Þ generate the 3 dimensional representations
and finally D12D23D31ð0; 0; 0Þ ¼ ð2; 1; 0Þ  ð1; 0; 2Þ þ ð2; 0; 1Þ  ð0; 2; 1Þ þ ð1; 2; 0Þþ
ð0; 1; 2Þ belongs to A32, as a subspace of A33, has weight 0 and is a one dimensional
representation (invariant). These are all the lowest weight vectors which can be
generated using the Hirota maps and it is easy to see that they give the complete
decomposition by summing over dimensions with multiplicities:
3 3 3 ¼ 7 52  33  1:
A42
Briefly, we have a 9 dimensional representation with lowest weight vector
ð0; 0; 0; 0Þ 2 A42; three 7 dimensional representations coming from D12, D13 and D14
acting on ð0; 0; 0; 0Þ 2 A41; six 5 dimensional representations from D212, D213, D214,
D12D13, D12D14 and D14D13; six 3 dimensional representations coming from D
2
12D34,
D12D
2
34, D13D
2
24, D
2
13D24, D14D
2
32 and D
2
14D32; and three 1 dimensional coming from
D212D
2
34, D
2
13D
2
24 and D
2
14D
2
32. This is precisely the decomposition,
3 3 3 3 ¼ 9 73  56  36  13:
One may do many such examples by hand, reinforcing the conjecture that
the Hirota map and the action of e construct all the irreducibles in the tensor
product.
4. Counting lowest weight vectors. If our conjecture is correct the lowest weight
vectors in Agn will come from polynomials in the Dij which, when applied to
ð0; 0; . . . ; 0Þ, never yield an integer greater than n. This is the same as seeking poly-
nomials over k in the differences xi  xj of g variables x1; . . . ; xg which involve no
power of any xi greater than n.
Let R be the polynomial ring k½x1; . . . ; xg, I the ideal of R generated by the set
fxnþ11 ; . . . ; xnþ1g g and J the ideal of R generated by the element x1 þ x2 þ . . .þ xg. We
have the short exact sequence:
I,!R! R=I: ð4Þ
Let  be the map,
HIROTA DERIVATIVES AND REPRESENTATION THEORY 3
 : R! R; xi 7!xi  1
g
ðx1 þ x2 þ . . .þ xgÞ
which has kernel J and let  be the projection
 : R! R=J:
This is precisely the polynomial ring in the differences of the xi. Now apply  to (4).
It is easy to see that we obtain the short exact sequence,
I
I \ J ,!
R
J
! ðR=J Þ
I=ðI \ J Þ
and, by the isomorphism theorems,
ðR=J Þ
I=ðI \ J Þ ffi
ðR=J Þ
ðIþ J Þ=J ffi
R
Iþ J
which is the homogeneous (finite dimensional) k-algebra
Kng ¼
R
ðxnþ11 ; . . . xnþ1g ; x1 þ x2 þ ::þ xgÞR
: ð5Þ
This homogeneous algebra is graded by isobaric weight
Kng ¼
Mng
p¼0
Kn;pg
and the k-dimension of each component is the number of lowest weight vectors
constructed by use of Hirota maps. (Note that for p larger than some value between
0 and ng these dimensions are zero.) Each comes with an irreducible representation
of dimension ng 2pþ 1 and so we have to show that
Xng
p¼0
ðng 2pþ 1Þ dimKn;pg ¼ ðnþ 1Þg ð6Þ
which is the dimension of Agn. This is the same as showing that
ðngþ 1Þkngð1Þ  2k 0ngð1Þ ¼ ðnþ 1Þg
where kngðsÞ is the Hilbert-Poincare´ series of Kng,
kngðsÞ ¼
Xng
p¼0
s p dimKn;pg
and prime denotes the derivative with respect to s.
The assertion is relatively easily verified in particular instances using algorithms
based on Gro¨bner bases implemented in software packages like MAPLE [7]. How-
ever, the proof is not so easily accomplished for general n and g.
4 CHRIS ATHORNE
5. The Hilbert-Poincare´ Series. This section constitutes the meat of the present
paper and is devoted to obtaining a closed form expression for the Hilbert-Poincare´
series, kngðsÞ, of Kng. The algebra is constructed as the kernel of a differential operator
acting on a simpler graded algebra. Let hngðsÞ denote the Poincare´-Hilbert series of
the homogeneous algebra R=I and let  denote the differential operator
@x1 þ @x2 þ . . . @xg .
Lemma 4.  maps R=I to itself with kernel R=ðIþ JÞ.
Proof. This is a straightforward consequence of the theory of partial differential
equations with constant coefficients.
Lemma 5. The isobaric weight gives a grading of R=I
R=I ¼
Mng
p¼0
Mn;pg
(every component of which is nontrivial) and  maps Mn;pg to M
n;p1
g :
Proof. The grading is obvious and since the element a ¼ ðx1 . . . xgÞn provides a
basis for Mn;ngg and 
nga ¼ ðngÞ! 6¼ 0 we see that all the components are non trivial.
Lemma 6. The Hilbert-Poincare´ series of R=I is
hngðsÞ ¼ ð1þ sþ s2 þ . . .þ snÞg
Proof. By induction on g:
Mn;pg ¼Mn;pg1  xgMn;p1g1  x2gMn;p2g2  . . . xngMn;png1
and hence
hngðsÞ ¼ hng1ðsÞ þ shng1 þ s2hng1 þ . . .þ snhng1
and it is straightforward that
hn2 ¼ 1þ sþ s2 þ . . .þ sn:
The idea now is to show that R=I actually carries a representation of sl2ðkÞ in
which Kng appears as the span of lowest weight vectors. We achieve this by means of
the following constructions very loosely analogous to the Hodge star and cobound-
ary operations in the theory of de Rham complexes.
Definition 7. Let a be a monomial element of R=I. We define a to be the
monomial with the property a  a ¼ ðx1x2 . . . xgÞn. Now extend  by k-linearity to
the general element of R=I.  thus maps Mn;pg to Mn;ngpg .
HIROTA DERIVATIVES AND REPRESENTATION THEORY 5
It is easily seen, by the symmetry of the Hilbert-Poincare´ series,
snghngð1=sÞ ¼ hngðsÞ
that  is an isomorphism of k-vector spaces.
Definition 8. With the map  : Mn;pg !Mn;p1g we associate a ‘‘dual’’ map
 : Mn;pg !Mn;pþ1g defined by
 ¼  ð7Þ
Lemma 9. The operators ,  and H ¼  make R=I into an sl2ðkÞ-module.
Proof. By considering the action of the operators on a monomial element it can
easily be seen that:
 ¼ nðx1 þ . . .þ xgÞ  x21@1  . . . x2g@g
H ¼ ng 2ðx1@1 þ . . .þ xg@gÞ
½H; ¼ 2
½H;  ¼ 2
Theorem 10. R=I decomposes into a direct sum of irreducible sl2ðkÞ-modules iso-
morphic to Ad of dimensions d ¼ ngþ 1; ng 1; ng 3; . . . with multiplicities
	d ¼ dimMn;ðdþng1Þ=2g  dimMn;ðdþngþ1Þ=2g :
Proof. This follows from standard representation theory [2]. The Mn;pg are the
eigenspaces of H with weights ng 2p. Each lowest weight vector, v with ðvÞ ¼ 0, of
weight ng 2p belongs to an irreducible representation isomorphic to Ad of dimen-
sion d ¼ 2p ngþ 1: The highest dimensional irreducible is associated with the
lowest weight vector ðx1 . . . xgÞn 2Mn;ngg and has dimension ngþ 1. Factoring out
this irreducible we are left with dimMn;ng1g  dimMn;ngg lowest weight vectors in
Mn;ng1g each generating irreducibles of dimension ng 1. Factoring out these and
continuing we arrive at the result.
A consequence of this is that, because of the symmetry between highest and
lowest weight vectors, we can easily construct the kernel of  as the span of the
highest weight vectors and this gives the result:
Theorem 11. The Hilbert-Poincare´ series of R=ðIþ J Þ is
kngðsÞ ¼ ½ð1 sÞhgnðsÞþ ð8Þ
where ½þ denotes the truncation to the terms with positive coefficients.
6 CHRIS ATHORNE
Finally, we see that the formula (6) for the total dimensions of all the irreducible
representations generated by the Hirota maps is just same as the formula for the
total dimension of all the irreducibles in R=I, that is the dimension of R=I. Hence:
Theorem 12.
ðngþ 1Þkngð1Þ  2k 0ngð1Þ ¼ hngð1Þ ¼ ðnþ 1Þg: ð9Þ
The approach above immediately generalizes to the g-fold tensor product of
sl2ðkÞ-modules of arbitrary dimensions:
Agn1;...;ng ¼ An1  An2  . . . Ang :
We therefore consider the homogeneous algebra
R=ðxn1þ11 ; . . . ; xngþ1g ÞR;
whose Hilbert-Poincare´ polynomial is
hn1;n2;...;ngðsÞ ¼
Yg
i¼1
ð1þ sþ . . .þ s niÞ
and which carries a sl2ðkÞ-representation given by the differential operators
 ¼ @1 þ . . . @g and  ¼ , where  is defined by its k-linear extension from its
action on monomials a, namely,
aðaÞ ¼ xn11 xn22 . . . xngg :
Explicitly,  is given by
 ¼ n1x1 þ . . .þ ngxg  x21@1  . . . x2g@g
and
H ¼ ½;  ¼ n1 þ . . .þ ng  2ðx1@1 . . . xg@gÞ:
,  and H are easily seen to satisfy the commutation relations of sl2ðkÞ. Again the
Hilbert-Poincare´ polynomial is ½ð1 sÞQgi¼1ð1þ sþ . . .þ sniÞþ and the entire
decomposition arises from Hirota maps, because the dimensions satisfy the formula
ðn1 þ . . .þ ng þ 1Þkð1Þ  2k0ð1Þ ¼
Yg
i¼1
ðni þ 1Þ:
6. Conclusion. We have presented a derivation of Hilbert-Poincare´ series for
certain homogeneous algebras generated by the generalized Hirota map and shown
thereby that such maps generate the entire decomposition of tensor products of
standard representations into irreducibles. This can be regarded as an extension of
the constructive methods of classical invariant theory [9].
HIROTA DERIVATIVES AND REPRESENTATION THEORY 7
There are two obvious directions for further work.
Firstly, given the importance of the Hirota derivative in integrable systems the-
ory, one may ask whether the above results indicate an ‘explanation’ for this enig-
matic phenomenon. There are, for instance, connections with the Moyal bracket of
quantum mechanics [10].
Secondly, to what extent can this simple and attractive algebraic structure be
generalised to other Lie algebras or to differential operators other than ; for
example, operators of second order?
Finally, I should like to thank Ken Brown for one or two critical ears.
REFERENCES
1. C. Athorne, Algebraic invariants and generalized Hirota derivatives, Phys. Lett. A
256 (1999), 20–24.
2. W. Fulton and J. Harris, Representation theory (Springer-Verlag, 1991).
3. B. Grammaticos, A. Ramani and J. Hietarinta, Multilinear operators: the natural
extension of Hirota’s bilinear formalism, Phys. Lett. A 190 (1994), 65–70.
4. D. Hilbert, Lectures on algebraic invariants (Cambridge University Press, 1993).
5. Ryogo Hirota, Bilinearization of soliton equations, J. Phys. Soc. Japan 51 (1982),
323–331.
6. S. V. Manakov, S. Novikov, L. P. Pitaevskii and V. E. Zakharov, Theory of solitons
(Plenum, 1984).
7. Maple V Release 5.00 (Waterloo Maple Inc., 1997).
8. J. J. C. Nimmo, Hirota’s method, in Soliton theory: a survey of results, Nonlinear
Science: Theory and Applications (Manchester University Press, Manchester, 1990), 75–96.
9. P. J. Olver, Classical invariant theory (Cambridge University Press, 1999).
10. I. A. B. Strachan, A geometry for multidimensional integrable systems, J. Geom.
Phys. 21 (1997), 255–278.
8 CHRIS ATHORNE


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