Fundamental Theorem Of Invariant Theory Research Articles

Quantized Weyl algebras, the double centralizer property, and a new first fundamental theorem for Uq(gln)

Let P:=Pm×n denote the quantized coordinate ring of the space of m × n matrices, equipped with natural actions of the quantized enveloping algebras Uq(glm) and Uq(gln) . Let L and R denote the images of Uq(glm) and Uq(gln) in End(P) , respectively. We define a q-analogue of the algebra of polynomial-coefficient differential operators inside End(P) , henceforth denoted by PD , and we prove that L∩PD and R∩PD are mutual centralizers inside PD . Using this, we establish a new First Fundamental theorem of invariant theory for Uq(gln) . We also compute explicit formulas in terms of q-determinants for generators of the algebras Lh∩PD and Rh∩PD , where Lh and Rh denote the images of the Cartan subalgebras of Uq(glm) and Uq(gln) in End(P) , respectively. Our algebra PD and the algebra Pol(Matm,n)q that is defined in (Shklyarov et al 2004 Int. J. Math. 15 855–94) are related by extension of scalars, but we give a new construction of PD using deformed twisted tensor products.

Standard monomials and invariant theory for arc spaces I: General linear group

This is the first in a series of papers on standard monomial theory and invariant theory of arc spaces. For any algebraically closed field [Formula: see text], we construct a standard monomial basis for the arc space of the determinantal variety over [Formula: see text]. As an application, we prove the arc space analogue of the first and second fundamental theorems of invariant theory for the general linear group.

An integral second fundamental theorem of invariant theory for partition algebras

We prove that the kernel of the action of the group algebra of the Weyl group acting on tensor space (via restriction of the action from the general linear group) is a cell ideal with respect to the alternating Murphy basis. This provides an analogue of the second fundamental theory of invariant theory for the partition algebra over an arbitrary commutative ring and proves that the centraliser algebras of the partition algebra are cellular. We also prove similar results for the half partition algebras.

The first and second fundamental theorems of invariant theory for the quantum general linear supergroup

We develop a non-commutative polynomial version of the invariant theory of the quantum general linear supergroup Uq(glm|n). A non-commutative Uq(glm|n)-module superalgebra Pr|sk|l is constructed, which is the quantum analogue of the supersymmetric algebra over Ck|l⊗Cm|n⊕Cr|s⊗(Cm|n)⁎. We analyse the structure of the subalgebra of Uq(glm|n)-invariants in Pr|sk|l by using a quantum super analogue of Howe duality.The subalgebra of Uq(glm|n)-invariants in Pr|sk|l is shown to be finitely generated. We determine its generators and establish a surjective superalgebra homomorphism from a braided supersymmetric algebra onto it. This establishes the first fundamental theorem of invariant theory for Uq(glm|n).We show that the above mentioned superalgebra homomorphism is an isomorphism if and only if m⩾min⁡{k,r} and n⩾min⁡{l,s}, and obtain a PBW basis for the subalgebra of invariants in this case. When the homomorphism is not injective, we give a representation theoretical description of the generating elements of the kernel. This way we obtain the relations obeyed by the generators of the subalgebra of invariants, producing the second fundamental theorem of invariant theory for Uq(glm|n).We consider the special case n=0 in greater detail, obtaining a complete treatment of the non-commutative polynomial version of the invariant theory of Uq(glm). In particular, the explicit SFT proved here is believed to be new. We also recover the FFT and SFT of invariant theory for the general linear superalgebra from the classical limit (i.e. q→1) of our results.

THE SECOND FUNDAMENTAL THEOREM OF INVARIANT THEORY FOR THE ORTHOSYMPLECTIC SUPERGROUP

The first fundamental theorem of invariant theory for the orthosymplectic supergroup scheme $\text{OSp}(m|2n)$ states that there is a full functor from the Brauer category with parameter $m-2n$ to the category of tensor representations of $\text{OSp}(m|2n)$. This has recently been proved using algebraic supergeometry to relate the problem to the invariant theory of the general linear supergroup. In this work, we use the same circle of ideas to prove the second fundamental theorem for the orthosymplectic supergroup. Specifically, we give a linear description of the kernel of the surjective homomorphism from the Brauer algebra to endomorphisms of tensor space, which commute with the orthosymplectic supergroup. The main result has a clear and succinct formulation in terms of Brauer diagrams. Our proof includes, as special cases, new proofs of the corresponding second fundamental theorems for the classical orthogonal and symplectic groups, as well as their quantum analogues, which are independent of the Capelli identities. The results of this paper have led to the result that the map from the Brauer algebra ${\mathcal{B}}_{r}(m-2n)$ to endomorphisms of $V^{\otimes r}$ is an isomorphism if and only if $r<(m+1)(n+1)$.

Ringel Duals of Brauer Algebras via Super Groups

AbstractWe prove that the Brauer algebra, for all parameters for which it is quasi-hereditary, is Ringel dual to a category of representations of the orthosymplectic super group. As a consequence we obtain new and algebraic proofs for some results on the fundamental theorems of invariant theory for this super group over the complex numbers and also extend them to some cases in positive characteristic. Our methods also apply to the walled Brauer algebra in which case we obtain a duality with the general linear super group, with similar applications.

Partition algebras Pk(n) with 2k>n and the fundamental theorems of invariant theory for the symmetric group Sn

Assume $\mathsf{M}_n$ is the $n$-dimensional permutation module for the symmetric group $\mathsf{S}_n$, and let $\mathsf{M}_n^{\otimes k}$ be its $k$-fold tensor power. The partition algebra $\mathsf{P}_k(n)$ maps surjectively onto the centralizer algebra $\mathsf{End}_{\mathsf{S}_n}(\mathsf{M}_n^{\otimes k})$ for all $k, n \in \mathbb{Z}_{\ge 1}$ and isomorphically when $n \ge 2k$. We describe the image of the surjection $\Phi_{k,n}:\mathsf{P}_k(n) \to \mathsf{End}_{\mathsf{S}_n}(\mathsf{M}_n^{\otimes k})$ explicitly in terms of the orbit basis of $\mathsf{P}_k(n)$ and show that when $2k > n$ the kernel of $\Phi_{k,n}$ is generated by a single essential idempotent $\mathsf{e}_{k,n}$, which is an orbit basis element. We obtain a presentation for $\mathsf{End}_{\mathsf{S}_n}(\mathsf{M}_n^{\otimes k})$ by imposing one additional relation, $\mathsf{e}_{k,n} = 0$, to the standard presentation of the partition algebra $\mathsf{P}_k(n)$ when $2k > n$. As a consequence, we obtain the fundamental theorems of invariant theory for the symmetric group $\mathsf{S}_n$. We show under the natural embedding of the partition algebra $\mathsf{P}_n(n)$ into $\mathsf{P}_k(n)$ for $k \ge n$ that the essential idempotent $\mathsf{e}_{n,n}$ generates the kernel of $\Phi_{k,n}$. Therefore, the relation $\mathsf{e}_{n,n} = 0$ can replace $\mathsf{e}_{k,n} = 0$ when $k \ge n$.

Tensor ideals, Deligne categories and invariant theory

We derive some tools for classifying tensor ideals in monoidal categories. We use these results to classify tensor ideals in Deligne’s universal categories  $${\underline{\mathrm{Rep}}} O_\delta ,{\underline{\mathrm{Rep}}} GL_\delta $$ and  $${\underline{\mathrm{Rep}}} P$$ . These results are then used to obtain new insight into the second fundamental theorem of invariant theory for the algebraic supergroups of types A, B, C, D, P. We also find new short proofs for the classification of tensor ideals in $${\underline{\mathrm{Rep}}} S_t$$ and in the category of tilting modules for  $${\text {SL}}_2(\Bbbk )$$ with $$\mathrm{char}(\Bbbk )>0$$ and for  $$U_q(\mathfrak {sl}_2)$$ with q a root of unity. In general, for a simple Lie algebra $$\mathfrak {g}$$ of type ADE, we show that the lattice of such tensor ideals for $$U_q(\mathfrak {g})$$ corresponds to the lattice of submodules in a parabolic Verma module for the corresponding affine Kac–Moody algebra.

Invariants of the special orthogonal group and an enhanced Brauer category

We first give a short intrinsic, diagrammatic proof of the First Fundamental Theorem of invariant theory (FFT) for the special orthogonal group $\text{SO}_m(\mathbb{C})$, given the FFT for $\text{O}_m(\mathbb{C})$. We then define, by means of a presentation with generators and relations, an enhanced Brauer category $\widetilde{\mathcal{B}}(m)$ by adding a single generator to the usual Brauer category $\mathcal{B}(m)$, together with four relations. We prove that our category $\widetilde{\mathcal{B}}(m)$ is actually (and remarkably) {\em equivalent} to the category of representations of $\text{SO}_m$ generated by the natural representation. The FFT for $\text{SO}_m$ amounts to the surjectivity of a certain functor $\mathcal{F}$ on $\text{Hom}$ spaces, while the Second Fundamental Theorem for $\text{SO}_m$ says simply that $\mathcal{F}$ is injective on $\text{Hom}$ spaces. This theorem provides a diagrammatic means of computing the dimensions of spaces of homomorphisms between tensor modules for $\text{SO}_m$ (for any $m$). These methods will be applied to the case of the orthosymplectic Lie algebras $\text{osp}(m|2n)$, where the super-Pfaffian enters, in a future work.

On the second fundamental theorem of invariant theory for the orthosymplectic supergroup

Let OSp(V) be the orthosymplectic supergroup on an orthosymplectic vector superspace V of superdimension (m|2n). Lehrer and Zhang showed that there is a surjective algebra homomorphism Frr:Br(m−2n)→EndOSp(V)(V⊗r), where Br(m−2n) is the Brauer algebra of degree r with parameter m−2n. The second fundamental theorem of invariant theory in this setting seeks to describe the kernel KerFrr of Frr as a 2-sided ideal of Br(m−2n). In this paper, we show that KerFrr≠0 if and only if r≥rc:=(m+1)(n+1), and give a basis and a dimension formula for KerFrr. We show that KerFrr as a 2-sided ideal of Br(m−2n) is generated by KerFrcrc for any r≥rc, and we provide an explicit set of generators for KerFrcrc. These generators coincide in the classical case with those obtained in recent papers of Lehrer and Zhang on the second fundamental theorem of invariant theory for the orthogonal and symplectic groups. As an application we obtain the necessary and sufficient conditions for the endomorphism algebra Endosp(V)(V⊗r) over the orthosymplectic Lie superalgebra osp(V) to be isomorphic to Br(m−2n).

Integration formulas for Brownian motion on classical compact Lie groups

Combinatorial formulas for the moments of the Brownian motion on classical compact Lie groups are obtained. These expressions are deformations of formulas of B. Collins and P. Śniady for moments of the Haar measure and yield a proof of the First Fundamental Theorem of invariant theory and of classical Schur–Weyl dualities based on stochastic calculus.

The first fundamental theorem of invariant theory for the orthosymplectic super group

We give a new proof, inspired by an argument of Atiyah, Bott and Patodi, of the first fundamental theorem of invariant theory for the orthosymplectic super group. We treat in a similar way the case of the periplectic super group. Lastly, the same method is used to explain the fact that Sergeev's super Pfaffian, an invariant for the special orthosymplectic super group, is polynomial.

On the existence of real R-matrices for virtual link invariants

We characterize the virtual link invariants that can be described as partition function of a real-valued R-matrix, by being weakly reflection positive. Weak reflection positivity is defined in terms of joining virtual link diagrams, which is a specialization of joining virtual link diagram tangles. Basic techniques are the first fundamental theorem of invariant theory, the Hanlon–Wales theorem on the decomposition of Brauer algebras, and the Procesi–Schwarz theorem on inequalities for closed orbits.

Cross products, invariants, and centralizers

An algebra V with a cross product × has dimension 3 or 7. In this work, we use 3-tangles to describe, and provide a basis for, the space of homomorphisms from V⊗n to V⊗m that are invariant under the action of the automorphism group Aut(V,×) of V, which is a special orthogonal group when dimV=3, and a simple algebraic group of type G2 when dimV=7. When m=n, this gives a graphical description of the centralizer algebra EndAut(V,×)(V⊗n), and therefore, also a graphical realization of the Aut(V,×)-invariants in V⊗2n equivalent to the First Fundamental Theorem of Invariant Theory. We show how the 3-dimensional simple Kaplansky Jordan superalgebra can be interpreted as a cross product (super)algebra and use 3-tangles to obtain a graphical description of the centralizers and invariants of the Kaplansky superalgebra relative to the action of the special orthosymplectic group.

The Structure of the Kac–Wang–Yan Algebra

The Lie algebra $${\mathcal{D}}$$ of regular differential operators on the circle has a universal central extension $${\hat{\mathcal{D}}}$$ . The invariant subalgebra $${\hat{\mathcal{D}}^+}$$ under an involution preserving the principal gradation was introduced by Kac, Wang, and Yan. The vacuum $${\hat{\mathcal{D}}^+}$$ -module with central charge $${c \in \mathbb{C}}$$ , and its irreducible quotient $${\mathcal{V}_c}$$ , possess vertex algebra structures, and $${\mathcal{V}_c}$$ has a nontrivial structure if and only if $${c \in \frac{1}{2}\mathbb{Z}}$$ . We show that for each integer $${n > 0}$$ , $${\mathcal{V}_{n/2}}$$ and $${\mathcal{V}_{-n}}$$ are $${\mathcal{W}}$$ -algebras of types $${\mathcal{W}(2, 4,\dots,2n)}$$ and $${\mathcal{W}(2, 4,\dots, 2n^2 + 4n)}$$ , respectively. These results are formal consequences of Weyl’s first and second fundamental theorems of invariant theory for the orthogonal group $${{\rm O}(n)}$$ and the symplectic group $${{\rm Sp}(2n)}$$ , respectively. Based on Sergeev’s theorems on the invariant theory of $${{\rm Osp}(1, 2n)}$$ we conjecture that $${\mathcal{V}_{-n+1/2}}$$ is of type $${\mathcal{W}(2, 4,\dots, 4n^2 + 8n + 2)}$$ , and we prove this for $${n = 1}$$ . As an application, we show that invariant subalgebras of $${\beta\gamma}$$ -systems and free fermion algebras under arbitrary reductive group actions are strongly finitely generated.

The Brauer category and invariant theory

A category of Brauer diagrams, analogous to Turaev's tangle category, is introduced, and a presentation of the category is given; specifically, we prove that seven relations among its four generating homomorphisms suffice to deduce all equations among the morphisms. Full tensor functors are constructed from this category to the category of tensor representations of the orthogonal group $O(V)$ or the symplectic group $Sp(V)$ over any field of characteristic zero. The first and second fundamental theorems of invariant theory for these classical groups are generalised to the category theoretic setting. The major outcome is that we obtain new presentations for the endomorphism algebras of the module $V^{\otimes r}$. These are obtained by appending to the standard presentation of the Brauer algebra of degree $r$ one additional relation. This relation stipulates the vanishing of an element of the Brauer algebra which is quasi-idempotent, and which we describe explicitly both in terms of diagrams and algebraically. In the symplectic case, if $\dim V = 2n$, the element is precisely the central idempotent in the Brauer subalgebra of degree $n + 1$, which corresponds to its trivial representation. Since this is the Brauer algebra of highest degree which is semisimple, our generator is an exact analogue for the Brauer algebra of the Jones idempotent of the Temperley-Lieb algebra. In the orthogonal case the additional relation is also a quasi-idempotent in the integral Brauer algebra. Both integral and quantum analogues of these results are given, the latter of which involve the BMW algebras.

The second fundamental theorem of invariant theory for the orthogonal group

Let $V=\C^n$ be endowed with an orthogonal form and $G=\Or(V)$ be the corresponding orthogonal group. Brauer showed in 1937 that there is a surjective homomorphism $\nu:B_r(n)\to\End_G(V^{\otimes r})$, where $B_r(n)$ is the $r$-string Brauer algebra with parameter $n$. However the kernel of $\nu$ has remained elusive. In this paper we show that, in analogy with the case of $\GL(V)$, for $r\geq n+1$, $\nu$ has kernel which is generated by a single idempotent element $E$, and we give a simple explicit formula for $E$. Using the theory of cellular algebras, we show how $E$ may be used to determine the multiplicities of the irreducible representations of $\Or(V)$ in $V^{\ot r}$. We also show how our results extend to the case where $\C$ is replaced by an appropriate field of positive characteristic, and comment on quantum analogues of our results.

Characterizing partition functions of the vertex model

We characterize which graph parameters are partition functions of a vertex model over an algebraically closed field of characteristic 0 (in the sense of [P. de la Harpe, V.F.R. Jones, Graph invariants related to statistical mechanical models: examples and problems, J. Combin. Theory Ser. B 57 (1993) 207–227]). We moreover characterize when the vertex model can be taken so that its moment matrix has finite rank. Basic instruments are the Nullstellensatz and the First and Second Fundamental Theorems of Invariant theory for the orthogonal group.

Invariant theory and the $\mathcal{W}_{1+\infty}$ algebra with negative integral central charge

The vertex algebra W_{1+\infty,c} with central charge c may be defined as a module over the universal central extension of the Lie algebra of differential operators on the circle. For an integer n\geq 1, it was conjectured in the physics literature that W_{1+\infty,-n} should have a minimal strong generating set consisting of n^2+2n elements. Using a free field realization of W_{1+\infty,-n} due to Kac-Radul, together with a deformed version of Weyl's first and second fundamental theorems of invariant theory for the standard representation of GL_n, we prove this conjecture. A consequence is that the irreducible, highest-weight representations of W_{1+\infty,-n} are parametrized by a closed subvariety of C^{n^2+2n}.

Geometric Complexity Theory II: Towards Explicit Obstructions for Embeddings among Class Varieties

In [K. D. Mulmuley and M. Sohoni, SIAM J. Comput., 31 (2001), pp. 496–526], henceforth referred to as Part I, we suggested an approach to the P vs. $NP$ and related lower bound problems in complexity theory through geometric invariant theory. In particular, it reduces the arithmetic (characteristic zero) version of the $NP \not \subseteq P$ conjecture to the problem of showing that a variety associated with the complexity class $NP$ cannot be embedded in a variety associated with the complexity class P. We shall call these class varieties associated with the complexity classes P and $NP$. This paper develops this approach further, reducing these lower bound problems—which are all nonexistence problems—to some existence problems: specifically to proving the existence of obstructions to such embeddings among class varieties. It gives two results towards explicit construction of such obstructions. The first result is a generalization of the Borel–Weil theorem to a class of orbit closures, which include class varieties. The second result is a weaker form of a conjectured analogue of the second fundamental theorem of invariant theory for the class variety associated with the complexity class $NC$. These results indicate that the fundamental lower bound problems in complexity theory are, in turn, intimately linked with explicit construction problems in algebraic geometry and representation theory. The results here were announced in [K. D. Mulmuley and M. Sohoni, in Advances in Algebra and Geometry (Hyderabad, $2001$), Hindustan Book Agency, New Delhi, India, 2003, pp. 239–261].