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 .