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International Journal of Mathematics and Mathematical Sciences
Volume 2010 (2010), Article ID 326247, 19 pages
Universal Verma Modules and the Misra-Miwa Fock Space
1Department of Mathematics and Statistics, University of Melbourne, Parkville VIC 3010, Australia
2Department of Mathematics, MIT, 77 Massachusetts Ave, Cambridge, MA 02139, USA
Received 7 March 2010; Accepted 5 November 2010
Academic Editor: Alistair Savage
Copyright © 2010 Arun Ram and Peter Tingley. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The Misra-Miwa -deformed Fock space is a representation of the quantized affine algebra . It has a standard basis indexed by partitions, and the nonzero matrix entries of the action of the Chevalley generators with respect to this basis are powers of . Partitions also index the polynomial Weyl modules for as tends to infinity. We explain how the powers of which appear in the Misra-Miwa Fock space also appear naturally in the context of Weyl modules. The main tool we use is the Shapovalov determinant for a universal Verma module.
Fock space is an infinite dimensional vector space which is a representation of several important algebras, as described in, for example, [1, Chapter 14]. Here we consider the charge zero part of Fock space, which we denote by , and its -deformation . The space has a standard -basis and . Following Hayashi , Misra and Miwa  define an action of the quantized universal enveloping algebra on . The only nonzero matrix elements of the Chevalley generators in terms of the standard basis occur when is obtained by adding a single -colored box to , and these are powers of .
We show that these powers of also appear naturally in the following context: partitions with at most parts index polynomial Weyl modules for the integral quantum group . Let be the standard dimensional representation of . If the matrix element is nonzero then, for sufficiently large , contains the highest weight vector of weight. There is a unique such highest weight vector which satisfies a certain triangularity condition with respect to an integral basis of . We show that the matrix element is equal to , where is the Shapovalov form and is the valuation at the cyclotomic polynomial .
Our proof is computational, making use of the Shapovalov determinant [4–6]. This is a formula for the determinant of the Shapovalov form on a weight space of a Verma module. The necessary computation is most easily done in terms of the universal Verma modules introduced in the classical case by Kashiwara  and studied in the quantum case by Kamita . The statement for Weyl modules is then a straightforward consequence.
Before beginning, let us discuss some related work. In , Kleshchev carefully analyzed the highest weight vectors in a Weyl module for and used this information to give modular branching rules for symmetric group representations. Brundan and Kleshchev  have explained that highest weight vectors in the restriction of a Weyl module to give information about highest weight vectors in a tensor product of a Weyl module with the standard -dimensional representation of . Our computations put a new twist on the analysis of the highest weight vectors in , as we study them in their “universal” versions and by the use of the Shapovalov determinant. Our techniques can be viewed as an application of the theory of Jantzen  as extended to the quantum case by Wiesner .
Brundan  generalized Kleshchev's  techniques and used this information to give modular branching rules for Hecke algebras. As discussed in [14, 15], these branching rules are reflected in the fundamental representation of and its crystal graph, recovering much of the structure of the Misra-Miwa Fock space. Using Hecke algebras at a root of unity, Ryom-Hansen  recovered the full action on Fock space. To complete the picture, one should construct a graded category, where multiplication by in the representation corresponds to a grading shift. Recent work of Brundan-Kleshchev  and Ariki  explains that one solution to this problem is through the representation theory of Khovanov-Lauda-Rouquier algebras [19, 20]. It would be interesting to explicitly describe the relationship between their category and the present work. Another related construction due to Brundan-Stroppel considers the case when the Fock space is replaced by , where is the natural module and are fixed natural numbers.
We would also like to mention very recent work of Peng Shan  which independently develops a similar story to the one presented here, but using representations of a quantum Schur algebra where we use representations of . The approach taken there is somewhat different and in particular relies on localization techniques of Beĭlinson and Bernstein .
This paper is arranged as follows. Sections 2 and 3 are background on the quantum group and the Fock space . Sections 4 and 5 explain universal Verma modules and the Shapovalov determinant. Section 6 contains the statement and proof of our main result relating Fock space and Weyl modules.
2. The Quantum Group and Its Integral Form
This is a very brief review, intended mainly to fix notation. With slight modifications, the construction in this section works in the generality of symmetrizable Kac-Moody algebras. See [23, Chapters 6 and 9] for details.
2.1. The Rational Quantum Group
is the associative algebra over the field of rational functions generated by with relations The algebra is a Hopf algebra with coproduct and antipode given by respectively, (see [23, Section 9.1]).
As a -vector space, has a triangular decomposition where the inverse isomorphism is given by multiplication (see [23, Proposition 9.1.3]). Here is the subalgebra generated by the for , is the subalgebra generated by the for , and is the subalgebra generated by the for .
2.2. The Integral Quantum Group
Let . For and , let in . The restricted integral form is the -subalgebra of generated by , , and for ,,. As discussed in [24, Section 6], this is an integral form in the sense that As with , the algebra has a triangular decomposition where the isomorphism is given by multiplication (see [23, Proposition 9.3.3]). In this case, is the subalgebra generated by the , is the subalgebra generated by the , and is generated by and for , , and .
2.3. Rational Representations
The Lie algebra of matrices has standard basis , where is the matrix with 1 in position and 0 everywhere else. Let . Let be the weight of given by . Define to be the set of integral weights, the set of dominant integral weights, the set of dominant polynomial weights, the set of positive roots, the root lattice, the positive part of the root lattice, and the negative part of the root lattice, respectively.
For an integral weight , the Verma module for of the highest weight is where is the one dimensional vector space over with action given by
Theorem 2.1 (see [23, Chapter 10.1]). If then has a unique finite dimensional quotient and the map is a bijection between , and the set of irreducible finite dimensional -modules.
A singular vector in a representation of is a vector such that for all i.
2.4. Integral Representations
3. Partitions and Fock Space
A partition is a finite length nonincreasing sequence of positive integers. Associated to a partition is its Ferrers diagram. We draw these diagrams as in Figure 1 so that, if , then is the number of boxes in row (rows run southeast to northwest ). Say that is contained in if the diagram for fits inside the diagram for and let be the collection of boxes of that are not in . For each box , the content is the horizontal position of and the color is the residue of modulo . In Figure 1, the numbers are listed below the diagram. The size of a partition is the total number of boxes in its Ferrers diagram.
The set of dominant polynomial weights from Section 2.3 is naturally identified with partitions with at most parts. If , then as -modules. The diagram of is obtained from the diagram of by adding a box on row , and appears in the sum on the right side of (3.1) if and only if is a partition. See, for example, [26, Section 6.1, Formula 6.8] for the classical statement and [23, Proposition 10.1.16] for the quantum case.
3.2. The Quantum Affine Algebra
Let be the quantized universal enveloping algebra corresponding to the -node Dynkin diagram
More precisely, is the algebra generated by , for , with relations See [23, Definition Proposition 9.1.1]. The algebra is the quantum group corresponding to the nontrivial central extension of the algebra of polynomial loops in .
3.3. Fock Space
Define -deformed Fock space to be the vector space with basis . Our is only the charge 0 part of Fock space described in . Fix and partitions such that is a single box. Define to be the set of addable boxes of color , the set of removable boxes of color , the left removable-addable difference, and the right removable-addable difference, respectively.
Theorem 3.1 (see [25, Theorem 10.6]). There is an action of on determined by where denotes the color of and the sum is over partitions which differ from by removing (resp. adding) a single -colored box.
As a -module, is isomorphic to an infinite direct sum of copies of the basic representation . Using the grading of where has degree , the highest weight vectors in occur in degrees divisible by , and the number of the highest weight vectors in degree is the number of partitions of . Then, , where has degree , and acts trivially on the second factor (see [27, Propositin 2.3]). Note that we are working with the “derived” quantum group , not the “full” quantum group , which is why there are no -shifts in the summands of .
Comment 1. Comparing with [25, Chapter 10], our is equal to Ariki's and our is equal to Ariki's . However, these numbers play a slightly different role in Ariki's work, which is explained by a different choice of conventions.
4. Universal Verma Modules
The purpose of this section is to construct a family of representations which are universal Verma modules in the sense that each can be “evaluated" to obtain any given Verma module. This notion was defined by Kashiwara  in the classical case and was studied in the quantum case by Kamita .
4.1. Rational Universal Verma Modules
Let . This field is isomorphic to the field of fractions of via the map For each , define a -linear automorphism by where is the inner product on defined by . Let be the one-dimensional vector space over with basis vector and action given by
The-shifted rational universal Verma module is the -module The universal Verma module is actually a module over , where is the field of fractions of . However, if we identify with using the map , the action of on is not by multiplication, but rather is twisted by the automorphism . It is to keep track of the difference between the action of and multiplication that we use different notation for the generators of and (i.e., versus ).
4.2. Integral Universal Verma Modules
The field contains an -subalgebra which is isomorphic to via the restriction of the map in (4.1). The integral universal Verma module is the -submodule of generated by . By restricting (4.4), where is the -submodule of spanned by . In particular, is a free -module.
Let be the map defined by where is the inner product on defined by . There is a surjective -module homomorphism “evaluation at ”
For fixed , the maps and extend to a map from the subspace of and , respectively, where no denominators evaluate to 0. Where it is clear we denote both these extended maps by .
Example 4.1. Computing the action of on and ,
4.4. Weight Decompositions
Let be a -module. For each , we define the -weight space of to be The universal Verma module is a -module, where the second factor acts as multiplication. The weight space if and only if with in the positive part of the root lattice. These nonzero weight spaces and the weight decomposition of can be described explicitly by Here, is defined using the grading of with .
4.5. Tensor Products
Let be a -module and a -module. The tensor product is a -module, where the first factor acts via the usual coproduct and the second factor acts by multiplication on . In the case when and both have weight space decompositions, the weight spaces of are
We also need the following.
Proposition 4.2. The tensor product of a universal Verma module with a Weyl module satisfies
Proof. Fix . In general, has a Verma filtration (see, e.g., [28, Theorem 2.2]) and if is dominant for all such that then which can be seen by, for instance, taking central characters. The proposition follows since this is true for a Zariski dense set of weights .
5. The Shapovalov Form and the Shapovalov Determinant
5.1. The Shapovalov Form
The Cartan involution is the -algebra anti-involution of defined by The map is also a coalgebra involution. An -contravariant form on a -module is a symmetric bilinear form such that
It follows by the same argument used in the classical case  that there is an -contravariant form (the Shapovalov form) on each Verma module and this is unique up to rescaling. The radical of is the maximal proper submodule of , so for all . In particular, descends to an -contravariant form on .
Since fixes , there is a well-defined notion of an -contravariant form on a module. In particular, the restriction of the Shapovalov form on to is -contravariant.
5.2. Universal Shapovalov Forms
There are surjective maps of -algebras and defined by and , for . Using the triangular decomposition (2.7), there is an -linear surjection The standard universal Shapovalov form is the -bilinear form defined by for all . Here, and are as in (4.1) and (4.2). Since for , the form is -contravariant. As with the usual Shapovalov form, distinct weight spaces are orthogonal, where weight spaces are defined as in Section 4.4.
Evaluation at gives an -valued -contravariant form on by The form can be extended by linearity to an -contravariant form on .
5.3. The Shapovalov Determinant
Let be a -module with a chosen -contravariant form. Let be an basis for the -weight space of . Let be the determinant of the form evaluated on the basis . Changing the basis changes the determinant by a unit in , and we sometimes write to mean the determinant calculated on an unspecified basis ( which is only defined up to multiplication by unit in ). The Shapovalov determinant is
Define the partition function by Then, for any , and implies that and .
Proposition 5.2. Fix with . Choose an -basis for . Consider the -bases for and for . Then .
Proof. For , The result follows by taking determinants.
5.4. Contravariant Forms on Tensor Products
If and are -modules with -contravariant forms and , define an -bilinear form by . Similarly, for a module with -bilinear -contravariant form , define a -bilinear form on by Since is a coalgebra involution (i.e., , for ), the forms and are -contravariant.
In the case when , evaluation of the -contravariant form at gives an -contravariant form : for and . As in Section 4.3, this form can be extended to the -submodule of the rational module where no denominators evaluate to zero.
6. The Misra-Miwa Formula for from Representation Theory
Let us prepare the setting for our main result (Theorem 6.1). Fix and a partition . Let be a positive integer greater than the number of parts of . All calculations below are in terms of representations of .(1)Let be the standard -dimensional module. Since , (3.1) implies where the sum is over those indices for which is a partition. For ease of notation, let .(2)Fix an -basis of where has weight and . Recursively, define singular weight vectors in by (i)(ii)for each , the submodule of generated by contains all weight vectors of of weight greater than or equal to . Thus, using (6.1), for each there is a one-dimensional space of singular vectors of weight in , and this is not contained in (since ). This implies that there unique singular vector of weight in where we recall that . (3)There is a unique -contravariant form on normalized so that and a unique -contravariant form on normalized so that . As in Section 5.4, define a -contravariant form on by . For each , define an element by
Theorem 6.1. The Misra-Miwa operators from Section 3.3 satisfy where is the box , is the color of box as in Figure 1, is the th cyclotomic polynomial in , and is the number of factors of in the numerator of minus the number of factors of in the denominator of .
The proof of Theorem 6.1 will occupy the rest of this section. We will first prove a similar statement, Proposition 6.6, where the role of the Weyl modules is played by the universal Verma modules from Section 4. For ease of notation, let denote the module from Section 4.2.
Definition 6.2. Recursively define singular weight vectors and elements for by(i),
(ii)since generates as a module, Proposition 4.2 implies that, for each , there is a unique singular vector in , where and the factor of acts by multiplication on .
Lemma 6.3. For any weight , up to multiplication by a power of , where, as in Section 5.3, is the determinant of the Shapovalov form evaluated on an -basis for the weight space of .
Comment 2. In order for Lemma 6.3 to hold as stated, for each , one must calculate the in the numerator and denominator with respect to the same -basis. The power of which appears depends on this choice of -bases.
Proof of Lemma 6.3. For each fix an -basis for . Consider the following three -bases for :
Let denote the determinant of calculated on , where is one of ,, or . Let denote calculated with respect to the basis .
By the definition of the -contravariant form on (see Section 4.5), For , is one dimensional and is a power of . Hence, up to multiplication by a power of , (6.7) simplifies to
Notice that is isomorphic to , and is the union of -bases for each of these submodules. For each , and each define an basis of by Using , where the last equality uses Proposition 5.2. Here, as in Section 5.3, is the Shapovalov determinant calculated with respect to the basis .
The change of basis from to is unitriangular and the change of basis from to is unitriangular. Thus, , and so the right sides of (6.8) and (6.10) are equal. The lemma follows from this equality by rearranging.
Lemma 6.4. Up to multiplication by a power of ,
Proof. Fix . Setting in Lemma 6.3 and applying Theorem 5.1 we see that, up to multiplication by a power of , where, for each , is a unit in . The value is 0 unless and . If , then acts as the identity on , so the corresponding factors in the numerator and denominator cancel. Hence, we need only consider factors on the right hand side where , , and . If , then , and hence , so on the left hand since we only need to consider those factors where . Up to multiplication by a power of , the expression reduces to The last two expressions are equal because they are each a product over pairs with , and the factors of have been dropped because they are powers of . Using the fact that and making the change of variables and on the right side, (6.13) becomes For , the lemma now follows by induction. For , the result simply says that , which we already know.
Proposition 6.5. Let be a partition. Let (resp. ) be the set of boxes which can be added to (resp. removed from) on rows with such that the result is still a partition. Let and let be as in Figure 1. Then, up to multiplication by a power of ,
Proof. For , let be the last box in row of . By Lemma 6.4, up to multiplication by a power of , where the last equality is a simple calculation from definitions. The denominator on the right side is never zero, and the numerator is zero exactly when , so that is no longer a partition. If for any , then there is cancellation, giving (6.15). See Figure 2.
Proposition 6.6. Let be as in Section 3.3. For any partition ,
Proof. By Proposition 6.5, if is not a partition. If is a partition, then where the notation is as in Section 3.3. Since is divisible by if and only if is divisible by , and is never divisible by . The result now follows from Proposition 6.5.
The authors thank M. Kashiwara, A. Kleshchev, T. Tanisaki, R. Virk, and B. Webster for helpful discussions. A. Ram was partly supported by the NSF Grant no. DMS-0353038 and the Australian Research Council Grants nos. DP0986774 and DP0879951. P. Tingley was partly supported by the Australia Research Council Grant no. DP0879951 and the NSF Grant no. DMS-0902649.
- V. G. Kac, Infinite-Dimensional Lie Algebras, Cambridge University Press, Cambridge, UK, 3rd edition, 1990.
- T. Hayashi, “-analogues of Clifford and Weyl algebras—spinor and oscillator representations of quantum enveloping algebras,” Communications in Mathematical Physics, vol. 127, no. 1, pp. 129–144, 1990.
- K. Misra and T. Miwa, “Crystal base for the basic representation of ,” Communications in Mathematical Physics, vol. 134, no. 1, pp. 79–88, 1990.
- N. N. Šapovalov, “A certain bilinear form on the universal enveloping algebra of a complex semisimple Lie algebra,” Funkcional'nyi Analiz i ego Priloženija, vol. 6, no. 4, pp. 65–70, 1972.
- C. de Concini and V. G. Kac, “Representations of quantum groups at roots of 1,” in Modern Quantum Field Theory (Bombay, 1990), pp. 333–335, World Scientific Publishing, River Edge, NJ, USA, 1991.
- S. Kumar and G. Letzter, “Shapovalov determinant for restricted and quantized restricted enveloping algebras,” Pacific Journal of Mathematics, vol. 179, no. 1, pp. 123–161, 1997.
- M. Kashiwara, “The universal Verma module and the -function,” in Algebraic Groups and Related Topics (Kyoto/Nagoya, 1983), vol. 6 of Advanced Studies in Pure Mathematics, pp. 67–81, North-Holland, Amsterdam, The Netherlands, 1985.
- A. Kamita, “The -functions for prehomogeneous spaces of commutative parabolic type and universal Verma modules II—quantum cases,” preprint.
- A. S. Kleshchev, “Branching rules for modular representations of symmetric groups. II,” Journal für die Reine und Angewandte Mathematik, vol. 459, pp. 163–212, 1995.
- J. Brundan and A. Kleshchev, “Some remarks on branching rules and tensor products for algebraic groups,” Journal of Algebra, vol. 217, no. 1, pp. 335–351, 1999.
- J. C. Jantzen, “Zur charakterformel gewisser darstellungen halbeinfacher gruppen und lie-algebren,” Mathematische Zeitschrift, vol. 140, pp. 127–149, 1974.
- E. Wiesner, Translation functors and the Shapovalov determinant, thesis, University of Wisconsin-Madison, Madison, Wis, USA, 2005.
- J. Brundan, “Modular branching rules and the Mullineux map for Hecke algebras of type ,” Proceedings of the London Mathematical Society, vol. 77, no. 3, pp. 551–581, 1998.
- S. Ariki, “On the decomposition numbers of the Hecke algebra of ,” Journal of Mathematics of Kyoto University, vol. 36, no. 4, pp. 789–808, 1996.
- A. Lascoux, B. Leclerc, and J.-Y. Thibon, “Hecke algebras at roots of unity and crystal bases of quantum affine algebras,” Communications in Mathematical Physics, vol. 181, no. 1, pp. 205–263, 1996.
- S. Ryom-Hansen, “Grading the translation functors in type A,” Journal of Algebra, vol. 274, no. 1, pp. 138–163, 2004.
- J. Brundan and A. Kleshchev, “Graded decomposition numbers for cyclotomic Hecke algebras,” Advances in Mathematics, vol. 222, no. 6, pp. 1883–1942, 2009.
- S. Ariki, “Graded -Schur algebras,” http://arxiv.org/abs/0903.3453.
- M. Khovanov and A. D. Lauda, “A diagrammatic approach to categorification of quantum groups. I,” Representation Theory, vol. 13, pp. 309–347, 2009.
- R. Rouquier, “2-Kac-Moody algebras,” http://arxiv.org/abs/0812.5023.
- P. Shan, “Graded decomposition matrices of -Schur algebras via Jantzen filtration,” http://arxiv.org/abs/1006.1545.
- A. Beĭlinson and J. Bernstein, “A proof of Jantzen conjectures,” in I. M. Gel'fand Seminar, vol. 16 of Advances in Soviet Mathematics, pp. 1–50, American Mathematical Society, Providence, RI, USA, 1993.
- V. Chari and A. Pressley, A Guide to Quantum Groups, Cambridge University Press, Cambridge, UK, 1995.
- G. Lusztig, “Finite-dimensional Hopf algebras arising from quantized universal enveloping algebra,” Journal of the American Mathematical Society, vol. 3, no. 1, pp. 257–296, 1990.
- S. Ariki, Representations of Quantum Algebras and Combinatorics of Young Tableaux, vol. 26 of University Lecture Series, American Mathematical Society, Providence, RI, USA, 2002.
- W. Fulton and J. Harris, Representation Theory. A First Course, Readings in Mathematic, vol. 129 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1991.
- M. Kashiwara, T. Miwa, and E. Stern, “Decomposition of -deformed Fock spaces,” Selecta Mathematica. New Series, vol. 1, no. 4, pp. 787–805, 1995.
- J. C. Jantzen, Moduln mit Einem Höchsten Gewicht, vol. 750 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1979.