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The Macmahon Master Theorem and Higher Sugawara Operators

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The Macmahon Master Theorem and Higher Sugawara Operators The MacMahon Master Theorem and higher Sugawara operators Alexander Molev joint work with Eric Ragoucy University of Sydney LAPTH, Annecy In the super case, zij are elements of a superalgebra and ¯ı¯+¯ık¯+¯k¯ [zij , zkl ] = [zkj , zil ](−1) for all i, j, k, l ∈ {1,..., m+n}, where ¯ı = 0 for i 6 m and ¯ı = 1 for i > m, deg zij = ¯ı + ¯. Manin matrices A matrix Z = [zij ] with entries in an algebra is a Manin matrix if [zij , zkl ] = [zkj , zil ] for all i, j, k, l ∈ {1,..., m}. Manin matrices A matrix Z = [zij ] with entries in an algebra is a Manin matrix if [zij , zkl ] = [zkj , zil ] for all i, j, k, l ∈ {1,..., m}. In the super case, zij are elements of a superalgebra and ¯ı¯+¯ık¯+¯k¯ [zij , zkl ] = [zkj , zil ](−1) for all i, j, k, l ∈ {1,..., m+n}, where ¯ı = 0 for i 6 m and ¯ı = 1 for i > m, deg zij = ¯ı + ¯. −1 −1 I Z = ∂u − E u , zij = ∂u δij − eij u , eij ∈ glm, −∂u −∂u I Z = e (u + E), zij = e u δij + eij . −∂u I Z = e T (u), T (u) is the generator matrix for the Yangian Y(glm|n). Examples of Manin matrices. I zij are elements of any (super)commutative (super)algebra. −∂u −∂u I Z = e (u + E), zij = e u δij + eij . −∂u I Z = e T (u), T (u) is the generator matrix for the Yangian Y(glm|n). Examples of Manin matrices. I zij are elements of any (super)commutative (super)algebra. −1 −1 I Z = ∂u − E u , zij = ∂u δij − eij u , eij ∈ glm, −∂u I Z = e T (u), T (u) is the generator matrix for the Yangian Y(glm|n). Examples of Manin matrices. I zij are elements of any (super)commutative (super)algebra. −1 −1 I Z = ∂u − E u , zij = ∂u δij − eij u , eij ∈ glm, −∂u −∂u I Z = e (u + E), zij = e u δij + eij . Examples of Manin matrices. I zij are elements of any (super)commutative (super)algebra. −1 −1 I Z = ∂u − E u , zij = ∂u δij − eij u , eij ∈ glm, −∂u −∂u I Z = e (u + E), zij = e u δij + eij . −∂u I Z = e T (u), T (u) is the generator matrix for the Yangian Y(glm|n). Regard the matrix Z = [ zij ] as the element m+n X ¯ı¯+¯ m|n Z = eij ⊗ zij (−1) ∈ End C ⊗ Mm|n. i,j=1 m|n Denote by Za the matrix Z in the a-th copy of End C in m|n m|n End C ⊗ ... ⊗ End C ⊗Mm|n. | {z } k MacMahon’s Master Theorem The right quantum superalgebra Mm|n is generated by elements zij , subject to the Manin matrix relations: ¯ı¯+¯ık¯+¯k¯ [zij , zkl ] = [zkj , zil ](−1) for all i, j, k, l ∈ {1,..., m+n}. m|n Denote by Za the matrix Z in the a-th copy of End C in m|n m|n End C ⊗ ... ⊗ End C ⊗Mm|n. | {z } k MacMahon’s Master Theorem The right quantum superalgebra Mm|n is generated by elements zij , subject to the Manin matrix relations: ¯ı¯+¯ık¯+¯k¯ [zij , zkl ] = [zkj , zil ](−1) for all i, j, k, l ∈ {1,..., m+n}. Regard the matrix Z = [ zij ] as the element m+n X ¯ı¯+¯ m|n Z = eij ⊗ zij (−1) ∈ End C ⊗ Mm|n. i,j=1 MacMahon’s Master Theorem The right quantum superalgebra Mm|n is generated by elements zij , subject to the Manin matrix relations: ¯ı¯+¯ık¯+¯k¯ [zij , zkl ] = [zkj , zil ](−1) for all i, j, k, l ∈ {1,..., m+n}. Regard the matrix Z = [ zij ] as the element m+n X ¯ı¯+¯ m|n Z = eij ⊗ zij (−1) ∈ End C ⊗ Mm|n. i,j=1 m|n Denote by Za the matrix Z in the a-th copy of End C in m|n m|n End C ⊗ ... ⊗ End C ⊗Mm|n. | {z } k Let Hk and Ak denote the respective images in the algebra m|n m|n End C ⊗ ... ⊗ End C ⊗Mm|n. | {z } k Set ∞ X Bos = 1 + str Hk Z1 ... Zk , k=1 ∞ X k Ferm = 1 + (−1) str Ak Z1 ... Zk , k=1 | taking supertrace with respect to all k copies of End Cm n. Introduce the normalized symmetrizer and antisymmetrizer 1 X 1 X σ ∈ [S ], sgn σ · σ ∈ [S ]. k! C k k! C k σ∈Sk σ∈Sk Set ∞ X Bos = 1 + str Hk Z1 ... Zk , k=1 ∞ X k Ferm = 1 + (−1) str Ak Z1 ... Zk , k=1 | taking supertrace with respect to all k copies of End Cm n. Introduce the normalized symmetrizer and antisymmetrizer 1 X 1 X σ ∈ [S ], sgn σ · σ ∈ [S ]. k! C k k! C k σ∈Sk σ∈Sk Let Hk and Ak denote the respective images in the algebra m|n m|n End C ⊗ ... ⊗ End C ⊗Mm|n. | {z } k Introduce the normalized symmetrizer and antisymmetrizer 1 X 1 X σ ∈ [S ], sgn σ · σ ∈ [S ]. k! C k k! C k σ∈Sk σ∈Sk Let Hk and Ak denote the respective images in the algebra m|n m|n End C ⊗ ... ⊗ End C ⊗Mm|n. | {z } k Set ∞ X Bos = 1 + str Hk Z1 ... Zk , k=1 ∞ X k Ferm = 1 + (−1) str Ak Z1 ... Zk , k=1 | taking supertrace with respect to all k copies of End Cm n. Remarks. Original proof in the even case n = 0: I Garoufalidis, Lê and Zeilberger, 2006. I Other proofs: Konvalinka and Pak, 2007, Foata and Han, 2007, Hai and Lorenz, 2007. I The proof in the super case relies on the matrix form (1 − P12)[Z1, Z2] = 0 of the defining relations of Mm|n. Theorem (MacMahon Master Theorem for Mm|n). Bos × Ferm = 1. I Other proofs: Konvalinka and Pak, 2007, Foata and Han, 2007, Hai and Lorenz, 2007. I The proof in the super case relies on the matrix form (1 − P12)[Z1, Z2] = 0 of the defining relations of Mm|n. Theorem (MacMahon Master Theorem for Mm|n). Bos × Ferm = 1. Remarks. Original proof in the even case n = 0: I Garoufalidis, Lê and Zeilberger, 2006. I The proof in the super case relies on the matrix form (1 − P12)[Z1, Z2] = 0 of the defining relations of Mm|n. Theorem (MacMahon Master Theorem for Mm|n). Bos × Ferm = 1. Remarks. Original proof in the even case n = 0: I Garoufalidis, Lê and Zeilberger, 2006. I Other proofs: Konvalinka and Pak, 2007, Foata and Han, 2007, Hai and Lorenz, 2007. Theorem (MacMahon Master Theorem for Mm|n). Bos × Ferm = 1. Remarks. Original proof in the even case n = 0: I Garoufalidis, Lê and Zeilberger, 2006. I Other proofs: Konvalinka and Pak, 2007, Foata and Han, 2007, Hai and Lorenz, 2007. I The proof in the super case relies on the matrix form (1 − P12)[Z1, Z2] = 0 of the defining relations of Mm|n. Define the Berezinian Ber Z by X Ber Z = sgn σ · zσ(1)1 ... zσ(m)m σ∈Sm X 0 0 × sgn τ · zm+1,m+τ(1) ... zm+n,m+τ(n). τ∈Sn In the supercommutative specialization, " AB # Ber = det A · det(D − CA−1B)−1. CD Noncommutative Berezinian −1 0 Suppose a Manin matrix Z is invertible, Z = [zij ]. In the supercommutative specialization, " AB # Ber = det A · det(D − CA−1B)−1. CD Noncommutative Berezinian −1 0 Suppose a Manin matrix Z is invertible, Z = [zij ]. Define the Berezinian Ber Z by X Ber Z = sgn σ · zσ(1)1 ... zσ(m)m σ∈Sm X 0 0 × sgn τ · zm+1,m+τ(1) ... zm+n,m+τ(n). τ∈Sn Noncommutative Berezinian −1 0 Suppose a Manin matrix Z is invertible, Z = [zij ]. Define the Berezinian Ber Z by X Ber Z = sgn σ · zσ(1)1 ... zσ(m)m σ∈Sm X 0 0 × sgn τ · zm+1,m+τ(1) ... zm+n,m+τ(n). τ∈Sn In the supercommutative specialization, " AB # Ber = det A · det(D − CA−1B)−1. CD ∞ −1 X k Ber (1 − u Z ) = u str Hk Z1 ... Zk . k=0 Moreover, we have the noncommutative Newton identities: ∞ −1 X k k+1 Ber (1 + u Z ) ∂u Ber (1 + u Z ) = (−u) str Z . k=0 Theorem. If Z is a Manin matrix, then ∞ X k Ber (1 + u Z ) = u str Ak Z1 ... Zk , k=0 Moreover, we have the noncommutative Newton identities: ∞ −1 X k k+1 Ber (1 + u Z ) ∂u Ber (1 + u Z ) = (−u) str Z . k=0 Theorem. If Z is a Manin matrix, then ∞ X k Ber (1 + u Z ) = u str Ak Z1 ... Zk , k=0 ∞ −1 X k Ber (1 − u Z ) = u str Hk Z1 ... Zk . k=0 Theorem. If Z is a Manin matrix, then ∞ X k Ber (1 + u Z ) = u str Ak Z1 ... Zk , k=0 ∞ −1 X k Ber (1 − u Z ) = u str Hk Z1 ... Zk . k=0 Moreover, we have the noncommutative Newton identities: ∞ −1 X k k+1 Ber (1 + u Z ) ∂u Ber (1 + u Z ) = (−u) str Z . k=0 with the commutation relations (¯ı+¯)(k¯+¯l) eij [r], ekl [s] = δkj ei l [r + s] − δi l ekj [r + s](−1) δ δ ¯ + K δ δ (−1)¯ı − ij kl (−1)¯ı+k r δ , kj i l m − n r,−s r the element K is even and central, and eij [r] := eij t .
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