Signatures of Multiplicity Spaces in Tensor Products of sl 2 and U q (sl 2 ) Representations, and Applications

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1 Signatures of Multiplicity Spaces in Tensor Products of sl and U (sl Representations, and Applications Shashwat Kishore Abstract We study multiplicity space signatures in tensor products of sl and U (sl representations and their applications. We completely classify definite multiplicity spaces for generic tensor products of sl Verma modules. This provides a classification of a family of unitary representations of a basic uantized uiver variety, one of the first such classifications for any uantized uiver variety. We use multiplicity space signatures to provide the first real critical point lower bound for generic sl master functions. As a corollary of this bound, we obtain a simple and asymptotically correct approximation for the number of real critical points of a generic sl master function. We obtain a formula for multiplicity space signatures in tensor products of finite dimensional simple U (sl representations. Our formula also gives multiplicity space signatures in generic tensor products of sl Verma modules and generic tensor products of real U (sl Verma modules. Our results have relations with knot theory, statistical mechanics, uantum physics, and geometric representation theory. Contents Introduction and results Preliminaries 5. Signatures in sl weight representations Quantum groups case Classification of definite multiplicity spaces 6 3. Case: p = Case: n = Case: p =, n = Case: p =, n = Case: p = 3, n = Case: p, n Proof of Theorem Critical point bound 9 4. Preliminaries for the proof of Theorem Proof of Theorem Preliminaries for the proof of Corollary Proof of Corollary

2 INTRODUCTION AND RESULTS 5 Signatures for the uantum group case 5. Preliminaries for the proof of Theorem Proof of Theorem Conclusion and further work 5 7 Acknowledgements 6 A Classification List 8 A. Definitions B Proof of Theorem 3. 9 B. Proof of Theorem 3. for p = B. Proof of Theorem 3. for p = B.3 Proof of Theorem 3. for p = C Proof of Theorem 3.3 D Proof of Theorem 3.4 D. Proof of Theorem 3.4 for λ < D. Proof of Theorem 3.4 for λ > E Proof of Theorem E. Possible exceptional multiplicity spaces when ϵ < E. Possible exceptional multiplicity spaces when < ϵ < E.3 Classification of exceptions F Proof of Theorem F. Proof of Theorem 3.6 for p = F. Proof of Theorem 3.6 for p, n p F.3 Proof of Theorem 3.6 for p = n F.4 Proof of Theorem 3.6 for p = n G Proofs of preliminary results in Section 4 3 G. Proof of Lemma G. Proof of Lemma G.3 Proof of Lemma G.4 Proof of Lemma Introduction and results Verma modules form an important class of infinite dimensional representations of classical Lie algebras and uantum groups. Any simple Verma module with real highest weight (and = if applicable carries a uniue invariant nondegenerate Hermitian form known as the Shapovalov form. The signature of the Shapovalov form is closely related to several topics in representation theory and topology, including the unitary representation theory of uantized uiver varieties, the critical points of master functions, and

3 INTRODUCTION AND RESULTS the behavior of the Jones polynomial of knot seuences on the unit circle. In 005, Yee used Kazhdan- Lustzig polynomials to explicitly compute the signatures of the Shapovalov forms of Verma modules []. This development has opened the door to studying signatures and applications arising from constructions based on Verma modules. For any complex λ / Z +, the sl Verma module M λ is the uniue simple highest weight representation of sl of weight λ. It has a basis {v i } i=0 defined by v i = F i v 0, on which E and H act by Ev i = i(λ i+v i and Hv i = (λ iv i. When λ is real and not in Z +, the Shapovalov form on M λ is nondegenerate and satisfies the adjointness conditions H = H, E = F, and F = E. Given a set of real Verma modules M λ, M λ,..., M λn, the tensor product n i= M λ i carries a Hermitian form eual to the product of the forms on each factor. A fundamental problem related to tensor products is the decomposition problem: Given a tensor product of representations, rewrite it as a direct sum. In the case of simple real sl Verma modules satisfying the condition that the sum of highest weights is not in Z +, this type of decomposition exists and is uniue. More precisely, we have n i= M λ i = m=0 M ( λ i m E m, where each level m multiplicity space E m has dimension ( m+n n and is isomorphic to the space Hom(M( λi m, i M λ i. Moreover, the Shapovalov forms on the Verma modules induce a natural Hermitian form (and therefore a signature on each multiplicity space. The direct sum over m of the product of the forms on M (λ i m and E m euals the form on the tensor product n i= M λ i. The goal of this paper is to study the multiplicity space signatures and applications arising from Verma module tensor products, in the context of the Lie algebra sl. Our main results are Theorems.,., and.3. In particular, we completely classify the definite multiplicity spaces in any generic tensor product of sl Verma modules. Theorem.. The classification list in Appendix A classifies all definite multiplicity spaces in any tensor product of generic sl Verma modules. Remark. The definite multiplicity space classification gives the definite weight subspace classification as a corollary. The classification of definite multiplicity spaces given by Theorem. is especially interesting because it provides a classification of a family of unitary representations of a basic uantized uiver variety, one of the first such classifications for any uantized uiver variety. The uantized uiver variety arises from the universal enveloping algebra reduction and uantum Hamiltonian reduction given by Bezrukavnikov and Losev []. We then use signatures of multiplicity spaces to study a problem in differential topology. There is a family of sl master functions that arises in the study of Knizhnik-Zamolodchikov euations and spin chain Gaudin models [3]. Each master function F z,λ,m : C m C associated to the Gaudin model is indexed by two seuences of n real parameters, denoted z = (z,..., z n and λ = (λ,..., λ n, and a positive integer m. For an m-tuple (t, t,..., t m C m, the master function F z,λ,m is defined as F z,λ,m (t, t,..., t m = Disc(Q i Q(z i λi, where Q C[x] is defined as Q(x = m i= (x t i and Disc(Q = i<j (t i t j is its discriminant. A critical point of F z,λ,m is said to be real if the corresponding Q-polynomial has all real coefficients. The computation of the number of real critical points (denoted N z,λ,m of an arbitrary master function F z,λ,m is an open and difficult problem. Recently, Mukhin and Tarasov [4] have shown that We call a set of reals generic if they are not nonnegative integers and their sum is not a nonnegative integer. We call a set of real Verma modules generic if their highest weights are generic. 3

4 INTRODUCTION AND RESULTS signatures of Hermitian forms can be used along with the Bethe ansatz method to compute lower bounds for the numbers of real solutions to topological problems. We study real critical points of the master function using an approach based on Mukhin and Tarasov s work, a Bethe ansatz setup due to Etingof, Frenkel, and Kirillov [5], and a Bethe vector characterization due to Feigin, Frenkel, and Rybnikov [6]. Our second main result, Theorem., uses multiplicity space signatures to give the first lower bound for the number of real critical points of a generic sl master function. Theorem.. For a generic sl master function F z,λ,m, we have sgn(e m N z,λ,m, where sgn(e m denotes the signature of the space E m in the decomposition of n i= M λ i. We use our Theorem. lower bound on N z,λ,m along with an upper bound on N z,λ,m given by Mukhin and Varchenko [7] to show that dim(e m = ( m+n n is a good approximation for Nz,λ,m for m large. Corollary.. For fixed generic λ and z seuences, we have lim m N z,λ,m ( m+n n =. Finally, we extend our work to uantum groups and compute signatures of Hermitian forms in - deformed multiplicity spaces. For generic 3 on the complex unit circle, the uantum enveloping algebra U (sl is the standard -deformation of the enveloping algebra of sl [8]. For each nonnegative integer a, there is a uniue (a + -dimensional simple representation of U (sl, denoted Ṽa, and this representation carries a Shapovalov form. Moreover, a tensor product of two simple finite dimensional representations Ṽ a Ṽb is a representation of U (sl. The tensor product carries a uniue invariant nondegenerate Hermitian form induced by the Drinfeld coboundary structure [9]. The form is induced as follows. There is a standard universal R-matrix, defined for our choice of coproduct as R = H H i 0 ( i (i [i]! F i E i, where [i] = i and [i]! = [][]...[i]. The matrix R of the Drinfeld coboundary structure is defined in terms of the R-matrix by R = R(R R /. The form on the tensor product Ṽa Ṽb is given by (v w, v w = i (a iv, v (b i w, w, where R = i a i b i and the pairings (v, v and (w, w are computed using the forms on Ṽa and Ṽb. This construction can be extended to a construction of a uniue Hermitian form on any tensor product of finite dimensional simple U (sl representations. The form on a tensor product is contravariant due to the R factor, so the tensor product can be decomposed with multiplicity spaces, each carrying its own induced form. Our third main result, Theorem.3, is a formula for the multiplicity space signatures in an arbitrary tensor product of finite dimensional simple U (sl representations. A convenient feature of our formula is that it is combinatorial, in that it has only additions, and no subtractions. Theorem.3. We have the decomposition n i= Ṽa i = m 0 Ṽ( i a i m Ẽm, where sgn(ẽm = n m +m + +m n =m j= [ ( + j+ k= sign a k j m j k= m k ( j k= a j j k= m k m j ( aj+ Remark. Theorem.3 lifts to an identical result for a tensor product of U (sl Verma modules of real highest weights. When =, the formula is still valid (even though is not generic and gives multiplicity space signatures for a tensor product of generic sl Verma modules. The paper is organized as follows. In Section, we cover definitions and background information. In Section 3, we prove the classification given by Theorem.. In Section 4, we prove the critical point bound We call a master function generic if its real parameters are generic. 3 We call a number on the complex unit circle generic if it is not a root of unity. m j ]. 4

5 PRELIMINARIES given by Theorem. and the approximation given by Corollary.. In Section 5, we prove the signature formula given by Theorem.3. Preliminaries. Signatures in sl weight representations A weight representation of sl is a representation which carries a uniue Hermitian form and is isomorphic to a direct sum of finite dimensional H-weight spaces, with the weights bounded from above. The Hermitian form on the weight representation must also satisfy the condition that the weight spaces are orthogonal and H = H, E = F, and F = E. For instance, the Verma modules are weight representations due to their Shapovalov forms. To any weight representation of sl, we can associate a signature character, which is a (possibly infinite sum of powers of the formal symbol e with coefficients in the ring Z[s]/(s. For a weight representation V, the signature character, denoted by ch s (V, is defined as α (a α + b α se α, where the sum is taken over all weights α. For the weight space of weight α, the number of orthogonal basis vectors which pair with themselves to give a positive real is a α, while the number of orthogonal basis vectors which pair with themselves to give a negative real is b α. Hence, the dimension of the α-weight space is a α + b α while the signature of the α-weight space is a α b α. The signature characters satisfy some nice properties for standard constructions involving weight representations. Namely, for any weight representations V i, their tensor product and direct sum are both weight representations with ch s ( i V i = i ch s(v i and ch s ( i V i = i ch s(v i. Since Verma modules are weight representations, we can study their tensor product decompositions using signature characters. Let λ, λ,..., λ n be generic reals. For simplicity, from here on out, we will denote β λi = ch s (M λi. j=0 Proposition.. We have ch s (M λi = β λi = sj e λ i j if λ i < 0, λi j=0 eλi j + j= λ i sj+ λi e λ j if λ i > 0. There is a uniue decomposition of signature characters n i= β λ i = m=0 (a m + sb m β λ m. This decomposition exactly encodes the tensor product decomposition n i= M λ i = m=0 M λ m E m, in the sense that each multiplicity space E m has dimension eual to ( m+n n = am + b m and signature eual to a m b m. Hence, determining when E m is definite amounts to determining when a m = 0 or b m = 0. The signature character interpretation of the definite multiplicity space classification problem is useful because it translates a representation theoretic problem into a more tractable combinatorial problem.. Quantum groups case For on the complex unit circle, the deformed algebra U (sl over C[,, ] is generated by E, F, K, K with defining relations KEK = E, KF K = F, and EF F E = K K. For each a, the uniue (a +-dimensional simple U (sl representation Ṽa is generated by a highest weight vector v 0. In a certain basis {v i } a i=0, the operators E, F, and K act by Ev i = [a i + ]v i, F v i = [i + ]v i+, and Kv i = a i v i, where [k] = k k. As in the classical case, the representation Ṽa carries a Shapovalov form. 5

6 3 CLASSIFICATION OF DEFINITE MULTIPLICITY SPACES Proposition.. Under the Shapovalov form (, and the normalization (v 0, v 0 =, we have (v i, v i = ( a a j= [j] i j= [j] a i j= [j]. From Hopf algebra theory, we know that any tensor product n i= Ṽa i can be made into a representation of U (sl under the iteration of the coproduct map defined by (E = E +K E, (F = F +F K, and (K ± = K ± K ±. Furthermore, the Shapovalov forms and the Drinfeld coboundary induce a Hermitan form on the representation n i= Ṽa i. This tensor product can be uniuely decomposed into a sum of finite dimensional simple U (sl representations tensored with multiplicity spaces, where each multiplicity space carries an induced Hermitian form. We outline the proof of the Theorem.3 signature formula for the induced forms of these multiplicity spaces in Section 5 i = 3 Classification of definite multiplicity spaces In this section we use signature characters to prove the Theorem. definite multiplicity space classification for the decomposition of an arbitrary tensor product M λ M λ M λn. Throughout this section, let λ = n i= λ i, and assume the λ i s are reals in decreasing order, with the first p of them positive (0 p n. Refer to the tensor product n i= M λ i as the original tensor. By the formula for signature characters, we have that the signatures of the multiplicity spaces depend only on the values λ, λ,..., λ n. Thus, we define the explicit type of the original tensor to be λ, λ,..., λ n and the implicit type to be λ,..., λ n. We first classify definite spaces in the p = 0, n =, and n = 3 cases, and then apply those results to solve the n 4 case. Then we tie together our results to prove the classification given by Theorem. 3. Case: p = 0 Take 0 > λ > > λ n. Theorem 3.. In the decomposition n M λi i= = M λ m E m, m=0 each multiplicity space E m is definite. The even-level spaces are positive definite and the odd-level spaces are negative definite. Proof. The proof of this classification is a standard counting argument. 3. Case: n = Take arbitrary λ > λ, and define a function g as in Appendix A.. Theorem 3.. In the decomposition M λ M λ = M λ m E m, m=0 6

7 3 CLASSIFICATION OF DEFINITE MULTIPLICITY SPACES 3.3 Case: p =, n = 3 every multiplicity space is definite. The level m space is positive definite if g(λ, λ, m is positive and negative definite if g(λ, λ, m is negative. Proof. See Appendix B. 3.3 Case: p =, n = 3 Take λ > 0 > λ λ 3. Theorem 3.3. In the decomposition M λ M λ M λ3 = M λ m E m, we have the following classification for definite spaces. The definite spaces are exactly those with levels 0 through max{0, λ }. The even-level spaces in this range are positive definite and the odd-level spaces in this range are negative definite. Proof. See Appendix C. m=0 3.4 Case: p =, n = 3 Take λ λ > 0 > λ 3. Theorem 3.4. In the decomposition M λ M λ M λ3 = M λ m E m, we have the following classification for definite spaces: Case. λ < 0 There are λ + definite spaces. They are positive definite and have levels 0 through λ. Case. λ > 0 and λ + λ There are λ λ definite spaces. They are positive definite and they have levels 0 and λ + through λ. Case 3. λ > 0 and λ + > λ Except for one exception, only the level 0 space is definite (it is positive definite. The exception is explicit type, 0, 0,, in which there is one other definite space: it is the level space and it is negative definite. Proof. See Appendix D. m=0 3.5 Case: p = 3, n = 3 Take λ λ λ 3 > 0. Theorem 3.5. In the decomposition M λ M λ M λ3 = M λ m E m, 7 m=0

8 3 CLASSIFICATION OF DEFINITE MULTIPLICITY SPACES 3.6 Case: p, n 4 we have the following classification for definite spaces. The spaces with level 0 through λ 3 are all positive definite. Except for the following exceptions, these are the only definite spaces. Exceptions: For d 0 and explicit type 3d, d, d, d, the level d + and d + spaces are positive definite. For d 0 and explicit type 3d +, d, d, d, the level d + and d + 3 spaces are negative definite. For d and explicit type 3d, d, d, d, the level d + space is positive definite. For d and explicit type 3d +, d, d, d, the level d + space is negative definite. For d and explicit type 3d, d, d, d, the level d space is positive definite. For d and explicit type 3d, d, d, d, the level d + space is negative definite. Proof. See Appendix E. 3.6 Case: p, n 4 Theorem 3.6. In the decomposition n i= M λi we have the following classification for definite spaces. Case. p = = M λ m E m, m=0 There are +max{0, λ } definite spaces. They have levels 0 through max{0, λ }. In this range, the even-level spaces are positive definite and the odd-level spaces are negative definite. Case. p n There is one definite space. It is the level 0 space and it is positive definite. Case 3. p = n Case 4. p = n If λ < 0, there are λ p + definite spaces. They are positive definite and have levels 0 through λ p. If λ > 0 and λ + λ p, then there are λ p λ definite spaces. They are positive definite and they have levels 0 and λ + through λ p. If λ > 0 and λ + > λ p, then there is one definite space. It is the level 0 space and it is positive definite. Except for the exceptions listed below, there are λ p + definite spaces. They have levels 0 through λ p and they are positive definite. Exceptions: For any p 4, explicit type 0, 0, 0,..., 0 has 3 definite spaces. They are the level 0,, spaces and they are positive definite. 8

9 4 CRITICAL POINT BOUND 3.7 Proof of Theorem. For any p 4, explicit type,, 0,..., 0 has 3 definite spaces. They are the level 0,, spaces and they are positive definite. For p = 4, explicit type 3, 0, 0, 0, 0 has 3 definite spaces. They are the level 0 and spaces (which are positive definite and the level 3 space (which is negative definite. For p = 4, explicit type 4,,,, has 4 definite spaces. They are the level 0,,,4 spaces and they are positive definite. Proof. See Appendix F. 3.7 Proof of Theorem. Refer to the classification list in Appendix A. Theorem 3. gives the classification for Case. Theorem 3. gives the classification for Case. Theorem 3.3 gives the classification for Case 3. Theorem 3.6 gives the classification for Case 4. Theorems 3.4 and 3.6 give the classification for Case 5. Theorems 3.5 and 3.6 give the classification for Case 6. 4 Critical point bound In this section we use the Bethe ansatz method to prove the lower bound on the number of real critical points of the master function given by Theorem.. Then we use the bound to derive the asymptotic approximation for the number of real critical points of the master function given by Corollary.. Throughout this section, fix a generic λ-seuence and a generic z-seuence, each in R n, and consider the master function F z,λ,m F z,λ,m(t, t,..., t m = (t i t j (t i z k λ k. i<j i,k 4. Preliminaries for the proof of Theorem. We begin with some definitions and preliminary results. Definition. For any X U(sl and any i with i n, define an operator X i on n j= M λ j by X i = X. }{{}}{{} i operators n i operators Definition. For each i and j with i j n, define the Casimir tensor Ω ij by Ω ij = E i F j + F i E j + H i H j. Definition. For each i with i n, define the Gaudin model Hamiltonian H i by H i = j i Definition. For any (t,..., t m C m, take the polynomial Q to be Q(x = m i= (x t i. Definition. For any complex t with t z i, define an operator Z(t = n i= H i t z i. Definition. For any complex t with t z i for each i, define an operator Y (t by Y (t = n i= Denote the highest weight vector of each M λi by v i, and denote v = i v i. Ω ij z i z j. Definition. For any complex critical point (s, s,..., s m of F z,λ,m, define a vector b Q by b Q = Y (s Y (s Y (s m v. F i t z i. We are now ready to begin with preliminary results. For the proofs of these preliminary results, we use arguments similar to those developed in [5]. 9

10 4 CRITICAL POINT BOUND 4. Proof of Theorem. Lemma 4.. The H i s commute with each other, and they are Hermitian under the induced Hermitian form on Hom(M ( i λi m, i M λ i. Proof. See Appendix G.. Lemma 4.. We have Eb Q = 0. Proof. See Appendix G.. Lemma 4.3. We have Proof. We write [Z(s a, Y (s b ] = s a s b (Y (s a Y (s b. n ( Fj (Y (s a Y (s b = F j s a s b s a s b s j= a z j s b z j n ( F j (s b s a = s a s b (s j= a z j (s b z j n ( F j = (s a z j (s b z j j= = [Z(s a, Y (s b ], where in the last line we have used the commutation relations [H j, F j ] = F j and [H i, F j ] = 0 (i j. Lemma 4.4. We have [H i, Y (s Y (s Y (s m ]v = λiq (z i Q(z i b Q. As a corollary of Lemma 4.4, we have that b Q is an eigenvector of each H i with eigenvalue λiq (z i ( λ i n λ j j i z i z j. (This is just the eigenvalue computed in Lemma 4.4 plus the H i -eigenvalue of v. Q(z i + Lemma 4.5 (known, see [6]. The joint eigenvalues of the operators H i each have multiplicity, and the eigenvectors of the form Y (s Y (s Y (s m v are all the joint eigenvectors up to scalars. Lemma 4.6. If the joint eigenvector b Q has real joint eigenvalue, then the critical point (s,..., s m is real. That is, the corresponding Q-polynomial has real coefficients. Proof. See Appendix G Proof of Theorem. We now tie together the results from Section 4. to prove the critical point bound. Since Eb Q = 0 and Hb Q = ( m+ i λ ib Q, we have that b Q corresponds uniuely to a vector in the space Hom(M ( i λi m, i M λ i, which is isomorphic to the multiplicity space E m in the decomposition of i M λ i. Note that the Gaudin Hamiltonians H i descend to commuting Hermitian operators on the space E m. Thus, each real eigenvector of E m gives rise to a real critical point of the master function F z,λ,m. Since the absolute value of the signature of E m is a lower bound for the number of real eigenvectors for the commuting Hermitian operators H i, it is also a lower bound for the number of real critical points. Hence sgn(e m N z,λ,m. 0

11 4 CRITICAL POINT BOUND 4.3 Preliminaries for the proof of Corollary. 4.3 Preliminaries for the proof of Corollary. Definition. Define β n for integer n by β n = e ϵ β n+ϵ for any 0 < ϵ <. This is well-defined. Definition. For any real µ, let βµ be β µ evaluated at s =. Definition. For each nonnegative integer n, define a polynomial V n by V n (x = + x + x + + x n. We state the following results without proofs. Lemma 4.7. For an integer n, we have βn β = en+ V n+ (e if n 0 β en+ if n < 0. Lemma 4.8. For any integer n and any negative integer n, we have e n β n = β n +n if n + n < 0. Also, if n + n 0, then e n β n can be written as a finite sum of signature characters. 4.4 Proof of Corollary. Recall that N z,λ,m denotes the number of real critical points of F z,λ,m. From Theorem., we know sgn(e m N m dim(e m = ( m+n n. We will show, for fixed generic λi s, that as m approaches infinity, the ratio sgn(e m ( m+n n approaches. This will prove Corollary.. The tensor product π = n i= M λ i has signature character eual to n i= β λ i. We need to examine the signatures of the multiplicity spaces E m in the decomposition of π, which amounts to examining the coefficients of β λ m in the decomposition of n i= β λ i. We have n i= β λ i = e n i= {λ i} = e n i= {λ i} n i= β λ i ( p β e λ i V λi (e i= = e n i= λi +{λi} (β n = e n i= λi +{λi} ( p V λi (e i= n i=p+ β e λ i ( ( ( p m + n β n m ( m V λi (e. n m=0 i=

12 5 SIGNATURES FOR THE QUANTUM GROUP CASE Combining the above computation with Lemma 4.8, we obtain that for all sufficiently large m, sgn(e m ( m = where T = p i= λ i and the c i s are defined by the polynomial identity T ( m + n i ( i c i, ( n i=0 p V λi (x = i= T c i x i. The RHS in ( is a polynomial in m of degree n. Its leading term is i=0 m n (n! (c 0 c + + ( T c T. We will show that this leading term is nonzero and compute it explicitly. We have T i=0 ( i c i = p i= V λ i (. But V λi ( is eual to + or for each i, so T i=0 ( i c i also euals + or. Therefore, we obtain that ± mn (n! is the leading term of a polynomial which euals sgn(e m ( m for all sufficiently large m. We also know that mn (n! is the leading term of the polynomial in m defined by the binomial coefficient ( m+n n as desired.. Hence lim m sgn(e m = ( m+n n 5 Signatures for the uantum group case In this section we prove the signature formula given by Theorem.3. Throughout this section, fix a generic on the complex unit circle and nonnegative integers a and b. We start by proving some preliminary results. 5. Preliminaries for the proof of Theorem.3 Fix highest weight vectors v 0 and w 0 in the simple representations Ṽa and Ṽb of U (sl, respectively. We know that the tensor product Ṽa Ṽb admits a Shapovalov form (this form is defined using an R-matrix. As a conseuence of the Clebsch-Gordan formula, the tensor product decomposes as Ṽ a Ṽb = min{a,b} m=0 Ṽ a+b m Ẽm, where each multiplicity space has dimension and an associated scalar. The sign of this scalar (+ or is the signature of the multiplicity space. For each subrepresentation Ṽa+b m Ṽa Ṽb, there is a uniue highest weight vector of the form v 0 w m + m c m,i v i w m i, i=

13 5 SIGNATURES FOR THE QUANTUM GROUP CASE 5. Preliminaries for the proof of Theorem.3 where c m,i are scalars. We will call the highest weight vector determined by these scalars the unit-normalized highest weight vector. Given the unit-normalized highest weight vector u of Ṽa+b m Ṽa Ṽb, define the R-normalized highest weight vector u of the twisted subrepresentation Ṽa+b m Ṽb Ṽa by u = T Ru, where T is the twist operator and R is the universal R-matrix R = H H n 0 ( n (n F n E n. [n]! In the next two lemmas we explicitly compute the scalars for the unit-normalized and R-normalized highest weight vectors. Lemma 5.. In the subrepresentation Ṽa+b m Ṽa Ṽb, the unit-normalized highest weight vector u is where Proof. We have c m,0 = by definition. Write m c m,i v i w m i, i=0 c m,i = ( i ai i +i u = ( b m+i i ( a i m c m,i v i w m i. Since u is a highest weight vector, it is killed by the operator E: 0 = (Eu = (E + K Eu = i=0. m c m,i Ev i w m i + c m,i Kv i Ew m i+. i= Next, Ev i = [a i + ] v i, Kv i = a i+ v i, Ew m i+ = [b m + i] w m i. Using the construction of tensor product bases, we obtain: [a i + ]c m,i v i w m i = c m,i a i+ [b m + i] v i w m i so c m,i = ( c m,i Solving this recursion with the initial condition c m,0 =, we get as desired. c m,i = ( i ai i +i [b m + i] [a i + ]. ( b m+i i ( a i 3

14 5 SIGNATURES FOR THE QUANTUM GROUP CASE 5. Proof of Theorem.3 Lemma 5.. In the twisted subrepresentation Ṽa+b m Ṽb Ṽb, the R-normalized highest weight vector u is m c m,i v i w m, i=0 where and c m,0 = ( m 0.5ab am bm+m m c m,i = c m,0 ( i bi i +i ( b m ( a m ( a m+i i ( b i. Proof. Since the R-normalized highest weight vector of the twisted subrepresentation Ṽa+b m is a scalar multiple of the unit-normalized highest weight vector of the twisted subrepresentation Ṽa+b m, it suffices to show that c m,0 is the right value. Using the operation of F, we get that m T Ru = a(b m/ w m v 0 + c m,i w i v m i. i=0 Hence, we have the euation c m,0 ( m bm m +m ( a m ( b m = 0.5ab am, from which the lemma follows. 5. Proof of Theorem.3 Consider the decomposition with multiplicity spaces which preserves the Shapovalov form Ṽ a Ṽb = min{a,b} m=0 Ṽ a+b m Ẽm. To prove Theorem.3, it suffices to show that the signature of each Ẽm is eual to the sign (+ or of ( a m. (Iterating the two-factors formula gives the expression in Theorem.3. Fix an m. ( b m ( a+b+ m m From the Clebsch-Gordan formula, we know that dim(ẽm =. Thus, the signature sgn(ẽm is eual to + or. From R-matrix theory, we know that the sgn(ẽm euals the sign of (u, u, where u is the unitnormalized highest weight vector of the subrepresentation Ṽa+b m and u is the R-normalized highest weight vector of the twisted subrepresentation Ṽa+b m. By Lemmas 5. and 5., we have u = m i=0 c m,iv i w m i and u = m i=0 c m,i w i v m i. From the orthogonality of the canonical bases under the Shapovalov 4

15 6 CONCLUSION AND FURTHER WORK form, we get that in (u, u, the c m,i term in u interacts only with the c m,m i term in u. We have (c m,i v i w m i, c m,m iw m i v i = c m,i c m,m i (v i, v i (w m i, w m i ( ( a b = c m,i c m,m i i m i Hence (u, u is (u, u = ( m c m,0 bm+m m = ( m c m,0 bm+m m+ai+bi mi+i ( b m+i ( a i i m i ( a ( b i m i ( = ( m c m,0 bm+m m b m + i (a+b+ mi i m ( b m + i (a+b+ mi i i=0 ( a i. m i Using the relation ( l = ( l ( l l l l and the uantum Vandermonde identity, we get: m ( ( (u, u = ( m c m,0 bm+m m b m + i a i (a+b+ mi i i=0 m i m ( ( = bm+m m m b m a c m,0 (a+b+ mi i i=0 m i ( m a b = c m,0 m ( a + b + m = c m,0 ( m m = 0.5ab+ma+mb m +m ( a+b+ m b m ( m Dropping the powers of and taking the signs of the binomial coefficients gives Theorem.3. ( a m. ( a i ( a i m i ( b m i. 6 Conclusion and further work In this paper, we studied tensor product decompositions of sl Verma modules and their connections to other problems. We completely classified definite multiplicity spaces in arbitrary tensor products, used multiplicity space signatures to prove a critical point bound for generic master functions, and explicitly computed multiplicity space signatures for tensor products of -deformed finite dimensional simple representations. There are many possible directions for future work. The existence of a Shapovalov form-respecting decomposition holds for tensor products of Verma modules over any Kac-Moody algebras. Thus, we can ask for a definite multiplicity space classification for any Kac-Moody algebra. We expect that the answers to these uestions will give classifications of unitary representation families for more uantized uiver varieties. In addition, it would be interesting to further explore our real critical point bound. Specifically, we are 5

16 REFERENCES interested in how tight this bound is. Based on computer tests, we have found that the bound is relatively N good for many λ and z seuences, in the sense that the ratio m sgn(e m is usually close to. When m is small, however, the bound breaks down at least in some special cases. For example, we have proven the following results for m = (the two-variable master function case. Theorem 6.. For any n > 3, there exists some λ-seuence of length n such that for any z-seuence, we have the strict ineuality sgn(e < N. Theorem 6.. For n an even suare, there exists some λ-seuence of length n such that for any z-seuence, we have sgn(e = 0 and N n n. It would be interesting to derive uantitative estimates, for general m, on how unusual it is for sgn(e m to be significantly less than N m. Finally, there are several interesting problems related to multiplicity space signatures for uantum groups. The problem of determining multiplicity space signatures in decompositions of tensor powers Ṽ n a arises in physical settings. For the a = case, the multiplicity space signatures have been computed. However, for a = and higher, these signatures have yet to be computed. It would also be interesting to study the relationship between multiplicity space signatures in tensor products of U (sl representations and real critical points of -deformed master functions. We expect that these signatures will give rise to a real critical point bound for the -deformed master function, as in the standard case. 7 Acknowledgements I thank Mr. Gus Lonergan of the Massachusetts Institute of Technology for his invaluable guidance throughout this work. I thank Professor Pavel Etingof of MIT for his helpful conversations and the suggestion of using Mukhin and Tarasov s Bethe ansatz. I express gratitude to Dr. Tanya Khovanova and Dr. John Rickert for editing this paper. Lastly, I thank the Center for Excellence in Education, Massachusetts Institute of Technology, Research Science Institute, MIT-PRIMES program, and my sponsors for giving me the opportunity to pursue this research. References [] W. Yee. The Signature of the Shapovalov Form on Irreducible Verma Modules. Representation Theory (Electronic Journal of the American Mathematical Society, 9 (005. [] R. Bezrukavnikov, I. Losev. Etingof Conjecture for Quantized Quiver Varities. arxiv, 04. [3] I. Scherbak, A. Varchenko. Critical Points of Functions, sl Representations, and Fuchsian Differential Euations with only Univalued Solutions. Moscow Mathematical Journal, 3, 003. [4] E. Mukhin, V. Tarasov. Lower bounds for numbers of real solutions in problems of Schubert calculus. Presented at Perspectives of Modern Complex Analysis, 04. [5] P. Etingof, I. Frenkel, A. Kirillov. Lectures on Representation Theory and Knizhnik-Zamolodchikov Euations. American Mathematical Society Bookstore,

17 REFERENCES REFERENCES [6] B. Feigin, E. Frenkel, L. Rybnikov. Opers with irregular singularity and spectra of the shift of argument subalgebra. Duke Mathematical Journal,, 007. [7] E. Mukhin, A. Varchenko. Critical Points of Master Functions and Flag Varieties. Communications in Contemporary Mathematics, 6, 004. [8] J. Jantzen. Lectures on Quantum Groups. American Mathematical Society Bookstore, 996. [9] V. Drinfeld. Quasi-Hopf Algebras. Leningrad Mathematical Journal,,

18 A CLASSIFICATION LIST A Classification List Consider an arbitrary Verma module tensor product π = M λ M λ M λn. Assume the λ i s are in decreasing order, with the first p of them positive (here 0 p n. Define λ = n i= λ i. Define the explicit type of the tensor product π to be the seuence λ, λ, λ,..., λ n. Define a function g as in Subsection A.. Then we have the following classification for definite multiplicity spaces in the decomposition of π: Classification List. In the decomposition of π, the level m multiplicity space has dimension ( m+n n for each m. We have the following classification for definite multiplicity spaces in the decomposition: Case. p = 0 Case. n = Every space is definite. The even-level spaces are positive definite and the odd-level spaces are negative definite. Every space is definite. The level m space is positive definite if g(λ, λ, m is positive and negative definite if g(λ, λ, m is negative. Case 3. p =, n 3 There are max{, λ+ } definite spaces. They have levels 0 through max{0, λ }. In this range, the even-level spaces are positive definite and the odd-level spaces are negative definite. Case 4. p n, n 4 There is one definite space. It is the level 0 space and it is positive definite. Case 5. p = n, n 3 Case 6. p = n, n 3 a. If λ < 0, there are λ p + definite spaces. They are positive definite and have levels 0 through λ p. b. If λ > 0 and λ + λ p, then there are λ p λ other definite spaces. They are positive definite and they have levels 0 and λ + through λ p. b. If λ > 0 and λ + > λ p, then except for one exception only the level 0 space is definite (it is positive definite. The exception is explicit type, 0, 0,, in which there is one other definite space: it is the level space and it is negative definite. The first λ p + spaces (levels 0 through λ p are positive definite. Except for the following exceptions, there are no other definite spaces. For explicit type 3d, d, d, d where d 0, the level d + and d + spaces are positive definite. For explicit type 3d +, d, d, d where d 0, the level d + and d + 3 spaces are negative definite. For explicit type 3d, d, d, d where d, the level d + space is positive definite. For explicit type 3d +, d, d, d where d, the level d + space is negative definite. For explicit type 3d, d, d, d where d, the level d space is positive definite. For explicit type 3d, d, d, d where d, the level d + space is negative definite. 8

19 B PROOF OF THEOREM 3. A. Definitions For explicit type 3, 0, 0, 0, 0, the level 3 space is negative definite. For explicit type 4,,,,, the level 4 space is positive definite. For explicit type 0, 0, 0,..., 0 where p 4, the level space is positive definite. For explicit type,, 0,..., 0 where p 4, the level space is positive definite. A. Definitions Define a function sign by sign(x = x/ x. Define a function g on a triple of two nonintegral reals x > x and a nonnegative integer k as follows: If 0 > x > x then If x > 0, x < 0, x + x < 0, then If x > 0, x < 0, x + x > 0 then g(x, x, k = ( k g(x, x, k = sign [ ( x ] k g(x, x, k = If x > x > 0, then ( k : 0 k x +x ( x +x : x+x k x+x+ ( x +x + x +x + +k : x +x + k x + x : x + x + k x ( x +k : x + k g(x, x, k = : 0 k x g(x, x x, k x : x k max( x, x g(x, x x, k x + x + x x + x : max( x, x + k x + x ( k x x : x + x + k B Proof of Theorem 3. Since every multiplicity space has dimension, every multiplicity space is definite. We just need to determine which spaces are positive definite and which are negative definite, which we do using the following result (Lemma B.: Definition. Let βµ be β µ evaluated at s =. Definition. Let sign be a function on the nonzero reals defined by sign(x = x/ x. Lemma B.. We have e µ = β µ sign(µβ µ + (sign( µ β µ µ. Proof. Write out expression on the right hand side for each of the intervals µ >, > µ > 0, and 0 > µ. We divide into cases based on the number of positive highest weights. 9

20 B PROOF OF THEOREM 3. B. Proof of Theorem 3. for p = 0 B. Proof of Theorem 3. for p = 0 This case is straightforward by the p = 0 classification already computed in Theorem 3.. B. Proof of Theorem 3. for p = We only treat the case when λ is even, as the λ odd case is almost identical. The decomposition for the λ + λ < 0 follows easily from the relation e t β µ = β t+µ for µ < 0, t + µ < 0. The other case is that λ + λ > 0. In this case, we have β λ β λ = (e λ + e λ + + e λ λ + β λ λ β λ = (e λ + e λ + + e λ λ β λ + β λ m ( m+ λ, m λ where in the last line we have used the p = 0 classification. Expanding the product (e λ + e λ + + e λ λ β λ, we obtain β λ β λ = e λ + e λ + + e λ λ + β λ m ( m+. m= λ + Carefully rewriting the powers of e as signature characters using Lemma B. gives the decomposition in Theorem 3.. B.3 Proof of Theorem 3. for p = For any positive real t, write L t = e t + e t + + e t t. We have β λ β λ = (L λ + β λ λ (L λ + β λ λ = β λ β λ λ + β λ β λ λ β λ λ β λ λ + L λ L λ, and λ L λ L λ = (m + e λ m + m=0 λ m= λ λ + λ + m= λ λ e λ m ( λ + λ m + e λ m. The classifications for p and n = give a closed form for each of β λ β λ λ, β λ β λ λ, and β λ λ β λ λ, as a sum of signature characters. Carefully rewriting the powers of e in L λ L λ signature characters using Lemma B. gives the decomposition in Theorem 3.. as 0

21 D PROOF OF THEOREM 3.4 C Proof of Theorem 3.3 Our strategy is to identify a finite range of possible definite spaces and examine the spaces in the range using characters. We start with a lemma. Lemma C.. No space with level more than λ is definite. Proof (Lemma C.. We address two cases, based on the sign of λ. Case. λ < 0 Case. λ > 0 We need to show that there are no definite spaces other than the level 0 positive definite space. Note that β λ β λ3 contains β λ+λ 3 + sβ λ+λ 3. Then from the n = classification, it is clear from the product β λ (β λ +λ 3 + sβ λ +λ 3, which is contained in β λ β λ β λ3, that no space other than the level 0 space is definite. From the n = classification, the product β λ β λ3 contains β λ λ β λ 3. Hence, β λ β λ β λ3 contains β λ β λ λ β λ 3. Since λ + λ λ + λ 3 < 0, the same argument as the previous subcase shows that no space with level more than λ is definite, as desired. From Lemma C., the only possible definite spaces are those with level between 0 and λ (inclusive. Upon developing the product β λ β λ β λ3 = β λ ( s m β λ+λ 3 m m=0 with repeated use of the n = classification, we obtain that the even-level spaces in our range are positive definite and the odd-level spaces in our range are negative definite. D Proof of Theorem 3.4 Our strategy is to identify a finite range of possible definite spaces and examine spaces in the range using characters. We divide into cases based on the sign of λ. D. Proof of Theorem 3.4 for λ < 0 Lemma D.. No space with level greater than λ is definite. Proof. Since λ + λ 3 < 0, the closed form for two tensors gives that the product of β λ and β λ3 contains the product of β λ λ and β λ3. Hence, β λ β λ β λ3 contains β λ β λ λ β λ3. By the partial closed form for three tensors we get the claim. Lemma D.. Every space with level at most λ is positive definite.

22 D PROOF OF THEOREM 3.4 D. Proof of Theorem 3.4 for λ > 0 Proof. From the closed form for two tensors, we obtain that in the decomposition of M λ M λ, the spaces with level at most λ are all positive definite. Applying the closed form for two tensors again, we get that if the following condition is satisfied for 0 k λ : λ + λ k λ + λ k λ + λ λ then the claim holds. This is euivalent to: λ + λ λ which is true. Proof of λ < 0 case of Theorem 3.4. Lemma D. shows that there are no definite spaces with level more than λ. Lemma D. shows that all spaces with level at most λ are positive definite. Thus, the λ < 0 case of Theorem 3.4 holds. D. Proof of Theorem 3.4 for λ > 0 From the closed form for two tensors, the product β λ β λ β λ3 contains the product of β λ λ β λ β λ3 if λ + λ 3 < λ, and it contains β λ λ + β λ β λ3 if λ + λ 3 = λ. From this we get that all definite spaces have levels at most λ +, so we only need to focus on this finite set of spaces. Our strategy here is to write β λ β λ β λ3 as a sum of powers of e and use Lemma B. (see Appendix B. to determine the definite spaces. We begin by writing out the e-decomposition for the first λ + powers of e in (β λ β λ β λ3 : Lemma D.3. If k λ, then the coefficient of e λ k in the (β λ β λ β λ3 is k+. If λ k > λ, then the coefficient of e λ k is k+ if (k λ and k+ (k λ otherwise. Proof. Use the fact that each term of e λ k in the final product arises from a sum of products of e λ i, e λ i, e λ 3 i 3 for i + i + i 3 = k. There are ( k+ such products, and we can write them all out. If k λ, these terms are easy to analyze. If k > λ, we must write out four cases based on the parities of k and λ. After writing down the answer in each case, it is easy to see that the description provided covers all cases. Corollary D.. Define a function r by r(i = 0 if i is even and r(i = if i is odd. Define a function t by t(i = if i positive and t(i = 0 if i nonpositive. Define a function f by f(λ, λ, k = k + (k λ r(k λ t(k λ if 0 k λ + and f(λ, λ, k = 0 if k is not in that range. Then an euivalent statement of Lemma D.3 is that the coefficient of e λ k in the (β λ β λ β λ3 is f(λ, λ, k for 0 k λ. Corollary D.. Using the same logic as in the proof of Lemma D.3, we get that the coefficient of e λ λ + in the (β λ β λ β λ3 is f(λ, λ, λ +. We now have an explicit description of the original signature character product written as a sum of powers of e. We will now find the definite spaces (which have levels k between 0 and λ +, tackling the k λ and k = λ + cases separately. We begin by solving the k λ case. Using Lemma B., the

23 D PROOF OF THEOREM 3.4 D. Proof of Theorem 3.4 for λ > 0 fact that the function x x x is cyclic, and a division of the interval [0, λ + ] into several intervals based on the signs of λ k and k + λ, we get the following result (Lemma D.4: Lemma D.4. We have the following results on the regular signatures of the first λ + spaces. (Checking positive/negative definiteness of each level k space is the same as checking when the regular signature is k + or k. If 0 k λ, then the signature of the level k space is f(λ, λ, k f(λ, λ, k. If λ+ k min( λ, λ, then the signature of the level k space is f(λ, λ, k + f(λ, λ, k f(λ, λ, λ k. If λ + k λ, then the signature of the level k space is f(λ, λ, k + f(λ, λ, k Lemma D.5. If k λ, then the level k space is positive definite iff λ + k λ. In particular if λ + > λ, then the only positive definite space in the first λ + spaces is the level 0 space. Proof. If λ + k λ, then Corollary D. and Lemma D.4 give that the level k space is definite. Now we show that no other λ k > 0 has level k space positive definite. If k λ, then using Lemma D.4, the maximum possible value for the signature of the level k space is + (k λ < k +. If λ+ k min(λ, λ, then we must address several cases. If k > λ, λ k λ, or λ k λ, then some bounding shows that the signature of the level k space is less than k +. The other possibility is that k > λ, λ k > λ, and λ k λ. In this case the signature of the level k space is eual to This level k space is positive definite when k + λ k + + λ k λ. λ k λ k + = λ. Viewing the LHS as a function of k, we see that as k increases by, the LHS decreases by. Evaluating the LHS at k = 0 and k = gives that the above euality holds only if k is eual to one of or λ λ λ λ. But k can t eual either of these as k λ+. So there are no positive definite spaces of level k when λ+ k min(λ, λ. 3

24 D PROOF OF THEOREM 3.4 D. Proof of Theorem 3.4 for λ > 0 If λ + k λ, then the maximum possible value of the signature of the level k space is k + (k λ < k +. So, the only positive definite spaces are the ones in the given range. Lemma D.6. There are no negative definite spaces with level at most λ. Proof. Consider a level k space with 0 k λ. If k λ, then the minimum possible signature of the level k space is: 0 (k λ > k If λ+ k min( λ, λ, then the minimum possible value of the signature of the level k space is (k + λ k + k + λ λ k + k λ k. If λ + k λ, then the minimum possible value of the signature of the level k space is k + (k λ > k. So there are no negative definite spaces. Lemmas D.5 and D.6 classify definite spaces with level at most λ. All that remains is to determine when the level λ + space is definite. Lemma D.7. The only tensors with the level λ + space definite are tensors with explicit type, 0, 0,. In this explicit type, the level space is negative definite. Proof. First we check when the λ + space is positive definite. As λ λ λ the maximum possible value of the signature of the level λ space is: f(λ, λ, λ + + f(λ, λ, λ λ < λ +. So this space can never be positive definite. For negative definiteness, we have that the minimum possible value of the signature is: f(λ, λ, λ + + f(λ, λ, λ f(λ, λ, λ λ, which is at least λ λ + λ. For this space to be negative definite, then, we must have λ λ λ + λ Rearranging and using λ λ + λ +, we get that for this space to be negative definite, we must have λ. Checking the (finitely many explicit types with this condition yields the given exception. 4

25 E PROOF OF THEOREM 3.5 Proof of λ > 0 subcase of Theorem 3.4. From initial observations, only spaces with level at most λ + can be definite. Lemmas D.5 and D.6 classify definite spaces with level at most λ, while Lemma D.7 shows when the level λ + is definite. Pulling this claims together gives the λ > 0 subcase of Theorem 3.4. Subsections D. and D. together prove the classification in Theorem 3.4. E Proof of Theorem 3.5 An easy corollary of the n = classification is that the first λ 3 + multiplicity spaces are indeed positive definite. Thus, we need only classify multiplicity spaces with level greater than λ 3, which we will call the exceptional multiplicity spaces. To do this, we first determine all possible exceptional spaces in Subsections E. and E.. Then in Subsection E.3 we use character computations to determine which of those instances actually produce exceptional spaces. We begin with some notation. Write A = λ, a = λ λ, and similarly define B, b and C, c for λ and λ 3 respectively. Define ϵ = a + b + c. We divide into subcases based on the floor of ϵ, which is between 0 and (inclusive. E. Possible exceptional multiplicity spaces when ϵ < Lemma E.. If the original tensor does not have explicit type 3d, d, d, d, 3d, d, d, d, 3d, d, d, d, or 3d +, d, d, d, then it has no definite spaces with level more than λ 3. If the original tensor has one of those types, then the possible exceptional definite spaces have levels d + (first type, d (second type, d + and d + (third type, and d + (fourth type. Proof. By fudging the fractional parts of λ, λ, λ 3, we can ensure that b + c < while maintaining the original tensor s explicit type. If this condition is satisfied, than by the closed form for two tensors, the signature character product β λ β λ β λ3 contains β λ β λ β λ3 λ 3. From the closed form of two positive and one negative Vermas, we obtain that the only spaces with level more than λ 3 that can be definite in the original tensors are spaces with level k, where k satisfies λ + λ + λ 3 λ λ 3 k λ + λ 3. This ineuality is euivalent to A + B C + ϵ + λ 3 k B + + λ 3. Using some case analysis and bounding, we obtain that the only possible exceptional definite spaces occur in types: Explicit type 3d, d, d, d, in which the level d + space could be definite. Explicit type 3d, d, d, d, in which the level d space could be definite. Explicit type 3d, d, d, d, in which the level d + and d + spaces could be definite. Explicit type 3d +, d, d, d, in which the level d + space could be definite. 5