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The weight multiplicities of finite dimensional simple Lie algebras can be computed individually using various methods. Still, it is hard to derive explicit closed formulas. Similarly, explicit closed formulas for the multiplicities of maximal weights of affine Kac–Moody algebras are not known in most cases. In this paper, we study weight multiplicities for both finite and affine cases of classical types for certain infinite families of highest weights modules. We introduce new classes of Young tableaux, called the

We prove that the quantum cluster algebra structure of a unipotent quantum coordinate ring A_q(\\mathfrak{n}(w)), associated with a symmetric Kac–Moody algebra and its Weyl group element w, admits a monoidal categorification via the representations of symmetric Khovanov–Lauda–Rouquier algebras. In order to achieve this goal, we give a formulation of monoidal categorifications of quantum cluster algebras and provide a criterion for a monoidal category of finite-dimensional graded R-modules to become a monoidal categorification, where R is a symmetric Khovanov–Lauda–Rouquier algebra. Roughly speaking, this criterion asserts that a quantum monoidal seed can be mutated successively in all the directions, once the first-step mutations are possible. Then, we show the existence of a quantum monoidal seed of A_q(\\mathfrak{n}(w)) which admits the first-step mutations in all the directions. As a consequence, we prove the conjecture that any cluster monomial is a member of the upper global basis up to a power of q^{1/2}. In the course of our investigation, we also give a proof of a conjecture of Leclerc on the product of upper global basis elements.

We introduce and investigate new invariants of pairs of modules$M$and$N$over quantum affine algebras$U_{q}^{\\prime }(\\mathfrak{g})$by analyzing their associated$R$-matrices. Using these new invariants, we provide a criterion for a monoidal category of finite-dimensional integrable$U_{q}^{\\prime }(\\mathfrak{g})$-modules to become a monoidal categorification of a cluster algebra.

The ( q ,  t )-Cartan matrix specialized at t = 1 , usually called the quantum Cartan matrix , has deep connections with (i) the representation theory of its untwisted quantum affine algebra, and (ii) quantum unipotent coordinate algebra, root system and quantum cluster algebra of skew-symmetric type . In this paper, we study the ( q ,  t )-Cartan matrix specialized at q = 1 , called the t - quantized Cartan matrix , and investigate the relations with (ii ′ ) its corresponding unipotent quantum coordinate algebra, root system and quantum cluster algebra of skew-symmetrizable type .

Let $U_q'({\\mathfrak {g}})$ be a quantum affine algebra with an indeterminate $q$, and let $\\mathscr {C}_{\\mathfrak {g}}$ be the category of finite-dimensional integrable $U_q'({\\mathfrak {g}})$-modules. We write $\\mathscr {C}_{\\mathfrak {g}}^0$ for the monoidal subcategory of $\\mathscr {C}_{\\mathfrak {g}}$ introduced by Hernandez and Leclerc. In this paper, we associate a simply laced finite-type root system to each quantum affine algebra $U_q'({\\mathfrak {g}})$ in a natural way and show that the block decompositions of $\\mathscr {C}_{\\mathfrak {g}}$ and $\\mathscr {C}_{\\mathfrak {g}}^0$ are parameterized by the lattices associated with the root system. We first define a certain abelian group $\\mathcal {W}$ (respectively $\\mathcal {W} _0$) arising from simple modules of $\\mathscr {C}_{\\mathfrak {g}}$ (respectively $\\mathscr {C}_{\\mathfrak {g}}^0$) by using the invariant $\\Lambda ^\\infty$ introduced in previous work by the authors. The groups $\\mathcal {W}$ and $\\mathcal {W} _0$ have subsets $\\Delta$ and $\\Delta _0$ determined by the fundamental representations in $\\mathscr {C}_{\\mathfrak {g}}$ and $\\mathscr {C}_{\\mathfrak {g}}^0$, respectively. We prove that the pair $( \\mathbb {R} \\otimes _\\mathbb {\\mspace {1mu}Z\\mspace {1mu}} \\mathcal {W} _0, \\Delta _0)$ is an irreducible simply laced root system of finite type and that the pair $( \\mathbb {R} \\otimes _\\mathbb {\\mspace {1mu}Z\\mspace {1mu}} \\mathcal {W} , \\Delta )$ is isomorphic to the direct sum of infinite copies of $( \\mathbb {R} \\otimes _\\mathbb {\\mspace {1mu}Z\\mspace {1mu}} \\mathcal {W} _0, \\Delta _0)$ as a root system.

Let U_q'(g) be a quantum affine algebra of arbitrary type and let C_g^0 be Hernandez-Leclerc’s category. We can associate the quantum affine Schur–Weyl duality functor F_D to a duality datum D in C_g^0 . In this paper, we introduce the notion of a strong (complete) duality datum D and prove that, when D is strong, the induced duality functor F_D sends simple modules to simple modules and preserves the invariants , and ^ınfty introduced by the authors. We next define the reflections S_k and S^-1_k acting on strong duality data D . We prove that if D is a strong (resp.\\ complete) duality datum, then S_k(D) and S_k^-1(D) are also strong (resp. complete) duality data. This allows us to make new strong (resp. complete) duality data by applying the reflections S_k and S^-1_k from known strong (resp. complete) duality data. We finally introduce the notion of affine cuspidal modules in C_g^0 by using the duality functor F_D , and develop the cuspidal module theory for quantum affine algebras similar to the quiver Hecke algebra case. When D is complete, we show that all simple modules in C_g^0 can be constructed as the heads of ordered tensor products of affine cuspidal modules. We further prove that the ordered tensor products of affine cuspidal modules have the unitriangularity property. This generalizes the classical simple module construction using ordered tensor products of fundamental modules.

With recent advancements in artificial intelligence technology, various studies are being conducted in the shipbuilding industry. Traditionally, hull form variation methods have relied on the intuition and expertise of designers, leading to inconsistent results and unintended changes in the ship’s main dimensions depending on the designer’s competence. Moreover, the iterative process of design variation and analysis to derive the optimal hull form is both costly and time-consuming. To address these issues, this study proposes an optimal hull design technique utilizing reinforcement learning, a type of unsupervised learning in machine learning. Reinforcement learning allows the model to learn from past policies by recording and accumulating the rewards associated with various actions taken by an agent in a specific environment. In this study, after calculating the main parameters of the ship, the agent defines a state representing hull information and performs local transformations of the bow and stern. The reward of reinforcement learning is defined as the change in total resistance due to the hull deformation, constrained by limiting the tolerance of the ship’s prismatic coefficient (CP) and longitudinal center of buoyancy. In this study, the problem is solved by comparing the proximal policy optimization algorithm and the deep deterministic policy gradient algorithm to find the best deep reinforcement learning model for the hull optimization problem. The results were compared with the genetic algorithm and speed-constrained multi-objective particle swarm optimization, and the optimal hull resistance values were less different, but the time of the reinforcement learning model was five times shorter. Graphical Abstract Graphical Abstract

We introduce the notions of quasi-Laurent and Laurent families of simple modules over quiver Hecke algebras of arbitrary symmetrizable types. We prove that such a family plays a similar role of a cluster in quantum cluster algebra theory and exhibits a quantum Laurent positivity phenomenon similar to the basis of the quantum unipotent coordinate ring $\\mathcal {A}_q(\\mathfrak {n}(w))$, coming from the categorification. Then we show that the families of simple modules categorifying Geiß–Leclerc–Schröer (GLS) clusters are Laurent families by using the Poincaré–Birkhoff–Witt (PBW) decomposition vector of a simple module $X$ and categorical interpretation of (co)degree of $[X]$. As applications of such $\\mathbb {Z}\\mspace {1mu}$-vectors, we define several skew-symmetric pairings on arbitrary pairs of simple modules, and investigate the relationships among the pairings and $\\Lambda$-invariants of $R$-matrices in the quiver Hecke algebra theory.

Let [g.sub.0] be a simple Lie algebra of type ADE and let [U'.sub.q](g) be the corresponding untwisted quantum affine algebra. We show that there exists an action of the braid group B([g.sub.0]) on the quantum Grothendieck ring [K.sub.t](g) of Hernandez-Leclerc's category [C.sup.0.sub.g]. Focused on the case of type [A.sub.N-1], we construct a family of monoidal autofunctors [[S.sub.i].sub.i[member of]z] on a localization [T.sub.N] of the category of finite-dimensional graded modules over the quiver Hecke algebra of type [A.sub.[infinity]]. Under an isomorphism between the Grothendieck ring K([T.sub.N]) of [T.sub.N] and the quantum Grothendieck ring [K.sub.t]([A.sup.(1).sub.N-1]), the functors [[S.sub.i].sub.1[less than or equal to]i[less than or equal to]N-1] recover the action of the braid group B([A.sub.N-1]). We investigate further properties of these functors. Key words: Quantum affine algebra; quantum Grothendieck ring; braid group action; quiver Hecke algebra; R-matrix.

We introduce a new family of real simple modules over the quantum affine algebras, called the affine determinantial modules, which contains the Kirillov-Reshetikhin (KR)-modules as a special subfamily, and then prove T-systems among them which generalize the T-systems among KR-modules and unipotent quantum minors in the quantum unipotent coordinate algebras simultaneously. We develop new combinatorial tools: admissible chains of i-boxes which produce commuting families of affine determinantial modules, and box moves which describe the T-system in a combinatorial way. Using these results, we prove that various module categories over the quantum affine algebras provide monoidal categorifications of cluster algebras. As special cases, Hernandez-Leclerc categories Cg0 and Cg− provide monoidal categorifications of the cluster algebras for an arbitrary quantum affine algebra.