Dongho Moon

MIXED TENSOR REPRESENTATIONS AND RATIONAL REPRESENTATIONS FOR THE GENERAL LINEAR LIE SUPERALGEBRAS

Let GLðrÞ denote the general linear group of the r r invertible complex matrices, V 1⁄4 C be the natural representation of GLðrÞ, and V be the f-fold tensor product representation of GLðrÞ. In [S1] and [S2] Schur studied the centralizer algebra, EndGLðrÞð VÞ, of GLðrÞ on V and constructed the polynomial representations of GLðrÞ as submodules of V. Let V be the dual of V, and ðV Þ q V be the mixed tensor representation of GLðrÞ consisting of p-fold tensor product of V and q-fold tensor product of V. [BCHLLS], [Ha], [KM], [Ko], and [St] studied the mixed tensor representation, the centralizer algebra EndGLðrÞð ðV Þ q VÞ, and the rational representations of GLðrÞ, and thereby they generalized results in [S1] and [S2]. For the complex general linear Lie superalgebra glðm; nÞ, it was shown in [BR1] that certain polynomial representations of glðm; nÞ can be constructed

Planar Rook Algebras and Tensor Representations of 𝔤𝔩(1 | 1)

We establish a connection between planar rook algebras and tensor representations of the natural two-dimensional representation of the general linear Lie superalgebra 𝔤𝔩(1 | 1). In particular, we show that the centralizer algebra is the planar rook algebra for all k ≥ 1, and we exhibit an explicit decomposition of into irreducible 𝔤𝔩(1 | 1)-modules. We obtain similar results for the quantum enveloping algebra and its natural two-dimensional module .

Tensor Product Representations of the Lie Superalgebra 𝔭(n) and Their Centralizers

Abstract We construct an associative algebra A k and show that there is a representation of A k on V ⊗k , where V is the natural 2n-dimensional representation of the Lie superalgebra 𝔭(n). We prove that A k is the full centralizer of 𝔭(n) on V ⊗k , thereby obtaining a “Schur-Weyl duality” for the Lie superalgebra 𝔭(n). This result is used to understand the representation theory of the Lie superalgebra 𝔭(n). In particular, using A k we decompose the tensor space V ⊗k , for k = 2 or 3, and show that V ⊗k is not completely reducible for any k ≥ 2.

A BIJECTIVE PROOF OF THE SECOND REDUCTION FORMULA FOR LITTLEWOOD-RICHARDSON COEFFICIENTS

There are two well known reduction formulae for structural constants of the cohomology ring of Grassmannians, i.e., LittlewoodRichardson coefficients. Two reduction formulae are a conjugate pair in the sense that indexing partitions of one formula are conjugate to those of the other formula. A nice bijective proof of the first reduction formula is given in the authors’ previous paper while a (combinatorial) proof for the second reduction formula in the paper depends on the identity between Littlewood-Richardson coefficients of conjugate shape. In this article, a direct bijective proof for the second reduction formula for Littlewood-Richardson coefficients is given. Our proof is independent of any previously known results (or bijections) on tableaux theory and supplements the arguments on bijective proofs of reduction formulae in the authors’ previous paper.

AN EXTENSION OF REDUCTION FORMULA FOR LITTLEWOOD-RICHARDSON COEFFICIENTS

There is a well-known classical reduction formula by Griffiths and Harris for Littlewood-Richardson coefficients, which reduces one part from each partition. In this article, we consider an extension of the reduction formula reducing two parts from each partition. This extension is a special case of the factorization theorem of Littlewood-Richardson coefficients by King, Tollu, and Toumazet (the KTT theorem). This case of the KTT factorization theorem is of particular interest, because, in this case, the KTT theorem is simply a reduction formula reducing two parts from each partition. A bijective proof using tableaux of this reduction formula is given in this paper while the KTT theorem is proved using hives.

Mixed Tensor Representations of Quantum Superalgebra q (gl(m, n))

Let 𝒢 = gl(m, n) denote the complex general linear Lie superalgebra, and 𝒰 q (𝒢) denote the quantized algebra of the universal enveloping algebra of 𝒢. Let V = C (q) m+n be the natural representation of 𝒰 q (𝒢) and V* be the dual representation of V. We study ⊗ a V ⊗ b V*, the mixed tensor representation of 𝒰 q (𝒢), consisting of a-copies of V and b-copies of V*. We prove that the centralizer algebra of 𝒰 q (𝒢) on ⊗ a V ⊗ b V*, End 𝒰 q (𝒢) (⊗ a V ⊗ b V*), is isomorphic to the 2-parameter Iwahori–Hecke algebra for m − n ≥ a + b. Using the commuting actions of 𝒰 q (𝒢) and , we construct maximal vectors of 𝒰 q (𝒢) in ⊗ a V ⊗ b V* whose weights are in one-to-one correspondence with the indexing set of the irreducible representations of .

KEY EXCHANGE PROTOCOL USING MATRIX ALGEBRAS AND ITS ANALYSIS

A key exchange protocol using commutative subalge- bras of a full matrix algebra is considered. The security of the proto- col depends on the di-culty of solving matrix equations XRY = T, with given matrices R and T. We give a polynomial time algorithm to solve XRY = T for the choice of certain types of subalgebras. We also compare the e-ciency of the protocol with the Di-e-Hellman key exchange protocol on the key computation time and the key size.