arXiv subjects

Mee Seong Im

Publications and source records attributed to Mee Seong Im.

At least 55 records · Page 3Linked to original sources

arXiv2019

Irreducible calibrated representations of periplectic Brauer algebras and hook representations of the symmetric group

We construct an infinite tower of irreducible calibrated representations of periplectic Brauer algebras on which the cup-cap generators act by nonzero matrices. As representations of the symmetric group, these are exterior powers of the standard representation (i.e. hook representations). Our approach uses the recently-defined degenerate affine periplectic Brauer algebra, which plays a role similar to that of the degenerate affine Hecke algebra in representation theory of the symmetric group. We write formulas for the representing matrices in the basis of Jucys--Murphy eigenvectors and we completely describe the spectrum of these representations. The tower formed by these representations provides a new, non-semisimple categorification of Pascal's triangle. Along the way, we also prove some basic results about calibrated representations of the degenerate affine periplectic Brauer algebra.

math.RT↗

arXiv2019

Grothendieck rings of periplectic Lie superalgebras

We describe explicitly the Grothendieck rings of finite-dimensional representations of the periplectic Lie superalgebras. In particular, the Grothendieck ring of the Lie supergroup $P(n)$ is isomorphic to the ring of symmetric polynomials in $x_1^{\pm 1}, \ldots, x_n^{\pm 1}$ whose evaluation $x_1=x_2^{-1}=t$ is independent of $t$.

math.RT↗

arXiv2019

On calibrated representations of the degenerate affine periplectic Brauer algebra

We initiate the representation theory of the degenerate affine periplectic Brauer algebra on $n$ strands by constructing its finite-dimensional calibrated representations when $n=2$. We show that any such representation that is indecomposable and does not factor through a representation of the degenerate affine Hecke algebra occurs as an extension of two semisimple representations with one-dimensional composition factors; and furthermore, we classify such representations with regular eigenvalues up to isomorphism.

math.RT↗

arXiv2019

Optimization problems with low SWaP tactical Computing

In a resource-constrained, contested environment, computing resources need to be aware of possible size, weight, and power (SWaP) restrictions. SWaP-aware computational efficiency depends upon optimization of computational resources and intelligent time versus efficiency tradeoffs in decision making. In this paper we address the complexity of various optimization strategies related to low SWaP computing. Due to these restrictions, only a small subset of less complicated and fast computable algorithms can be used for tactical, adaptive computing.

cs.AI↗

arXiv2018

Nonstandard approach to Hausdorff measure theory and an analysis of some sets of dimension less than $1$

We study various measure theories using the classical approach and then compute the Hausdorff dimension of some simple objects and self-similar fractals. We then develop a nonstandard approach to these measure theories and examine the Hausdorff measure in more detail. We choose to study Hausdorff measure over any other measures since it is well-defined for all sets, and widely used in many different areas in mathematics, physics, probability theory, and so forth. Finally we generate a particular set and compute its upper and lower Hausdorff dimension. We compare our set with box-counting dimension and other well-known fractal behaviors to analyze the set in a greater detail.

math.LO↗

arXiv2018

On the module structure of the center of hyperelliptic Krichever-Novikov algebras II

Let $R := R_{2}(p)=\mathbb{C}[t^{\pm 1}, u : u^2 = t(t-α_1)\cdots (t-α_{2n})] $ be the coordinate ring of a nonsingular hyperelliptic curve and let $\mathfrak{g}\otimes R$ be the corresponding current Lie algebra. \color{black} Here $\mathfrak g$ is a finite dimensional simple Lie algebra defined over $\mathbb C$ and \begin{equation*} p(t)= t(t-α_1)\cdots (t-α_{2n})=\sum_{k=1}^{2n+1}a_kt^k. \end{equation*} In earlier work, Cox and Im gave a generator and relations description of the universal central extension of $\mathfrak{g}\otimes R$ in terms of certain families of polynomials $P_{k,i}$ and $Q_{k,i}$ and they described how the center $Ω_R/dR$ of this universal central extension decomposes into a direct sum of irreducible representations when the automorphism group was the cyclic group $C_{2k}$ or the dihedral group $D_{2k}$. We give examples of $2n$-tuples $(α_1,\dots,α_{2n})$, which are the automorphism groups $\mathbb G_n=\text{Dic}_{n}$, $\mathbb U_n\cong D_n$ ($n$ odd), or $\mathbb U_n$ ($n$ even) of the hyperelliptic curves \begin{equation} S=\mathbb{C}[t, u: u^2 = t(t-α_1)\cdots (t-α_{2n})] \end{equation} given in [CGLZ17]. In the work below, we describe this decomposition when the automorphism group is $\mathbb U_n=D_n$, where $n$ is odd.

math.RT↗

arXiv2018

Families of orthogonal Laurent polynomials, hyperelliptic Lie algebras and elliptic integrals

We describe a family of polynomials discovered via a particular recursion relation, which have connections to Chebyshev polynomials of the first and the second kind, and the polynomial version of Pell's equation. Many of their properties are listed in Section 3. We show that these families of polynomials in the variable $t$ satisfy certain second order linear differential equations that may be of interest to mathematicians in conformal field theory and number theory. We also prove that these families of polynomials in the setting of Date-Jimbo-Kashiwara-Miwa algebras when multiplied by a suitable power of $t$ are orthogonal with respect to explicitly-described kernels. Particular cases lead to new identities of elliptic integrals (see Section 5).

math-ph↗

arXiv2018

On the module structure of the center of hyperelliptic Krichever-Novikov algebras

We consider the coordinate ring of a hyperelliptic curve and let $\mathfrak{g}\otimes R$ be the corresponding current Lie algebra where $\mathfrak g$ is a finite dimensional simple Lie algebra defined over $\mathbb C$. We give a generator and relations description of the universal central extension of $\mathfrak{g}\otimes R$ in terms of certain families of polynomials $P_{k,i}$ and $Q_{k,i}$ and describe how the center $Ω_R/dR$ decomposes into a direct sum of irreducible representations when the automorphism group is $C_{2k}$ or $D_{2k}$.

math.RT↗

arXiv2018

Generalized iterated wreath products of cyclic groups and rooted trees correspondence

Consider the generalized iterated wreath product $\mathbb{Z}_{r_1}\wr \mathbb{Z}_{r_2}\wr \ldots \wr \mathbb{Z}_{r_k}$ where $r_i \in \mathbb{N}$. We prove that the irreducible representations for this class of groups are indexed by a certain type of rooted trees. This provides a Bratteli diagram for the generalized iterated wreath product, a simple recursion formula for the number of irreducible representations, and a strategy to calculate the dimension of each irreducible representation. We calculate explicitly fast Fourier transforms (FFT) for this class of groups, giving literature's fastest FFT upper bound estimate.

math.RT↗

arXiv2018

Generalized iterated wreath products of symmetric groups and generalized rooted trees correspondence

Consider the generalized iterated wreath product $S_{r_1}\wr \ldots \wr S_{r_k}$ of symmetric groups. We give a complete description of the traversal for the generalized iterated wreath product. We also prove an existence of a bijection between the equivalence classes of ordinary irreducible representations of the generalized iterated wreath product and orbits of labels on certain rooted trees. We find a recursion for the number of these labels and the degrees of irreducible representations of the generalized iterated wreath product. Finally, we give rough upper bound estimates for fast Fourier transforms.

math.RT↗

arXiv2018

Complexity and mission computability of adaptive computing systems

There is a subset of computational problems that are computable in polynomial time for which an existing algorithm may not complete due to a lack of high performance technology on a mission field. We define a subclass of deterministic polynomial time complexity class called mission class, as many polynomial problems are not computable in mission time. By focusing on such subclass of languages in the context for successful military applications, we also discuss their computational and communicational constraints. We investigate feasible (non)linear models that will minimize energy and maximize memory, efficiency, and computational power, and also provide an approximate solution obtained within a pre-determined length of computation time using limited resources so that an optimal solution to a language could be determined.

math.OC↗

arXiv2018

Beta super-functions on super-Grassmannians

Israel M. Gelfand gave a geometric interpretation for general hypergeometric functions as sections of the tautological bundle over a complex Grassmannian $G_{k,n}$. In particular, the beta function can be understood in terms of $G_{2,3}$. In this manuscript, we construct one of the simplest generalizations of the Euler beta function by adding arbitrary-many odd variables to the classical setting. We also relate the beta super-function to the gamma and the hypergeometric function.

math-ph↗

arXiv2018

Categorification of Verma modules and indecomposable projective modules in the category $\mathcal I_{\mathfrak{g}}(\mathfrak{sl}_2)$ for $\mathfrak{sl}_2$

We categorify Verma and indecomposable projective modules in the category $\mathcal I_{\mathfrak{g}}(\mathfrak{sl}_2)$ for $\mathfrak{sl}_2$ using a tensor product decomposition theorem of T. J. Enright and work of J. Chuang and R. Rouquier, A. Licata and A. Savage (Hecke algebras, finite general linear groups, and Heisenberg categorification) and M. Khovanov (Heisenberg algebra and a graphical calculus).

math.RT↗

arXiv2018

The regular semisimple locus of the affine quotient of the cotangent bundle of the Grothendieck-Springer resolution

Let $G= GL_n(\mathbb{C})$, the general linear group over the complex numbers, and let $B$ be the set of invertible upper triangular matrices in $G$. Let $\mathfrak{b}=\text{Lie}(B)$. For $μ:T^*(\mathfrak{b}\times \mathbb{C}^n)\rightarrow \mathfrak{b}^*$, where $\mathfrak{b}^*\cong \mathfrak{g}/\mathfrak{u}$ and $\mathfrak{u}$ being strictly upper triangular matrices in $\mathfrak{g}=\text{Lie}(G)$, we prove that the Hamiltonian reduction $μ^{-1}(0)^{rss}/\!\!/B$ of the extended regular semisimple locus $\mathfrak{b}^{rss}$ of the Borel subalgebra is smooth, affine, reduced, and scheme-theoretically isomorphic to a dense open locus of $\mathbb{C}^{2n}$. We also show that the $B$-invariant functions on the regular semisimple locus of the Hamiltonian reduction of $\mathfrak{b}\times \mathbb{C}^n$ arise as the trace of a certain product of matrices.

math.AG↗

arXiv2018

Optimization and synchronization of programmable quantum communication channels

Quantum applications transmit and receive data through quantum and classical communication channels. Channel capacity, the distance and the photon path between transmitting and receiving parties and the speed of the computation links play an essential role in timely synchronization and delivery of information using classical and quantum channels. In this study, we analyze and optimize the parameters of the communication channels needed for the quantum application to successfully operate. We also develop algorithms for synchronizing data delivery on classical and quantum channels.

quant-ph↗

arXiv2018

The affine VW supercategory

We define the affine VW supercategory $\mathit{s}\hspace{-0.7mm}\bigvee\mkern-15mu\bigvee$, which arises from studying the action of the periplectic Lie superalgebra $\mathfrak{p}(n)$ on the tensor product $M\otimes V^{\otimes a}$ of an arbitrary representation $M$ with several copies of the vector representation $V$ of $\mathfrak{p}(n)$. It plays a role analogous to that of the degenerate affine Hecke algebras in the context of representations of the general linear group; the main obstacle was the lack of a quadratic Casimir element in $\mathfrak{p}(n)\otimes \mathfrak{p}(n)$. When $M$ is the trivial representation, the action factors through the Brauer supercategory $\mathit{s}\mathcal{B}\mathit{r}$. Our main result is an explicit basis theorem for the morphism spaces of $\mathit{s}\hspace{-0.7mm}\bigvee\mkern-15mu\bigvee$ and, as a consequence, of $\mathit{s}\mathcal{B}\mathit{r}$. The proof utilises the close connection with the representation theory of $\mathfrak{p}(n)$. As an application we explicitly describe the centre of all endomorphism algebras, and show that it behaves well under the passage to the associated graded and under deformation.

math.RT↗

arXiv2016

Unipotent invariants of filtered representations of quivers and the isospectral Hilbert scheme

Given any finite quiver, we consider a complete flag of vector spaces over each vertex. Consider the unipotent invariant subalgebra of the coordinate ring of the filtered quiver representation subspace. We prove that the dimension of the algebraic variety of the unipotent invariant subalgebra is finite. We also construct an ADHM analog for the Borel subalgebra setting, showing its birationality to the isospectral Hilbert scheme. Quiver-graded Steinberg varieties, quantum Hamiltonian reduction, and deformation quantization constructions for the nonreductive setting are discussed, ending with open problems.

math.AG↗

arXiv2016

On semi-invariants of filtered representations of quivers and the cotangent bundle of the enhanced Grothendieck-Springer resolution

We introduce the notion of filtered representations of quivers, which is related to usual quiver representations, but is a systematic generalization of conjugacy classes of $n\times n$ matrices to (block) upper triangular matrices up to conjugation by invertible (block) upper triangular matrices. With this notion in mind, we describe the ring of invariant polynomials for interesting families of quivers, namely, finite $ADE$-Dynkin quivers and affine type $\widetilde{A}$-Dynkin quivers. We then study their relation to an important and fundamental object in representation theory called the Grothendieck-Springer resolution, and we conclude by stating several conjectures, suggesting further research.

math.AG↗