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We continue the study of realization of the prefundamental modules $L_{r,a}^{\pm}$, introduced by Hernandez and Jimbo, in terms of unipotent quantum coordinate rings as in [J-Kwon-Park, Int. Math. Res. Not., 2023]. We show that the ordinary character of $L_{r,a}^{\pm}$ is equal to that of the unipotent quantum coordinate ring $U_q^-(w_r)$ associated to fundamental $r$-th coweight. When $r$ is cominuscule, we prove that there exists a $U_q(\mathfrak{b})$-module structure on $U_q^-(w_r)$, which is isomorphic to $L_{r,a\eta_r}^\pm$ for some $\eta_r \in \mathbb{C}^\times$., Comment: 36 pages

We show that there exists a unique crystal base of a parabolic Verma module over a quantum orthosymplectic superalgebra, which is induced from a $q$-analogue of a polynomial representation of a general linear Lie superalgebra., Comment: 43 pages

As a continuation of \cite{JLO1}, we investigate the quantum virtual Grothendieck ring $\frakK_q(\g)$ associated with a finite dimensional simple Lie algebra $\g$, especially of non-simply-laced type. We establish an isomorphism $\Uppsi_Q$ between the heart subring $\frakK_{q,Q}(\g)$ of $\frakK_q(\g)$ associated with a Dynkin quiver $Q$ of type $\g$ and the unipotent quantum coordinate algebra $\calA_q(\n)$ of type $\g$. This isomorphism and the categorification theory via quiver Hecke algebras enable us to obtain a presentation of $\frakK_q(\g)$, which reveals that $\frakK_q(\g)$ can be understood as a boson-extension of $\calA_q(\n)$. Then we show that the automorphisms, arising from the reflections on Dynkin quivers and the isomorphisms $\Uppsi_Q$, preserve the canonical basis $\sfL_q$ of $\frakK_q(\g)$. Finally, we prove that such automorphisms produce a braid group $B_\g$ action on $\frakK_q(\g)$.

The quantum Grothendieck ring of a certain category of finite-dimensional modules over a quantum loop algebra associated with a complex finite-dimensional simple Lie algebra $\mathfrak{g}$ has a quantum cluster algebra structure of skew-symmetric type. Partly motivated by a search of a ring corresponding to a quantum cluster algebra of {\em skew-symmetrizable} type, the quantum {\em virtual} Grothendieck ring, denoted by $\mathfrak{K}_q(\mathfrak{g})$, is recently introduced by Kashiwara--Oh \cite{KO23} as a subring of the quantum torus based on the $(q,t)$-Cartan matrix specialized at $q=1$. In this paper, we prove that $\mathfrak{K}_q(\mathfrak{g})$ indeed has a quantum cluster algebra structure of skew-symmetrizable type. This task essentially involves constructing distinguished bases of $\mathfrak{K}_q(\mathfrak{g})$ that will be used to make cluster variables and generalizing the quantum $T$-system associated with Kirillov--Reshetikhin modules to establish a quantum exchange relation of cluster variables. Furthermore, these distinguished bases naturally fit into the paradigm of Kazhdan--Lusztig theory and our study of these bases leads to some conjectures on quantum positivity and $q$-commutativity., Comment: We corrected typos in Theorem 5.2 and Proposition 5.25

In this paper, we investigate the behavior of monomials in the $q$-characters of the fundamental modules over a quantum affine algebra of untwisted type C. As a result, we give simple closed formulae for the $q$-characters of the fundamental modules in terms of sequences of vertices in $\mathbb{R}^2$, so-called paths, with an admissible condition. This may be viewed as a type C analog of the path description of $q$-characters in types A and B due to Mukhin--Young., Comment: v4: minor revision; v3: 23 pages, revised introduction and abstract, reference added, comments added on Remark 2.1, 3.1, 3.2, 4.5, 4.9, 4.10, added Section 4.2 to explain connected components, added Examples 2.4, 4.4, added Lemma 4.6, revised proof of Theorem 4.7, and several typos corrected; v2: 15 pages, typos corrected, reference corrected

We construct a crystal base of $U_q(\mathfrak{gl}(m|n))^-$, the negative half of the quantum superalgebra $U_q(\mathfrak{gl}(m|n))$. We give a combinatorial description of the associated crystal $\mathscr{B}_{m|n}(\infty)$, which is equal to the limit of the crystals of the ($q$-deformed) Kac modules $K(\lambda)$. We also construct a crystal base of a parabolic Verma module $X(\lambda)$ associated with the subalgebra $U_q(\mathfrak{gl}_{0|n})$, and show that it is compatible with the crystal base of $U_q(\mathfrak{gl}(m|n))^-$ and the Kac module $K(\lambda)$ under the canonical embedding and projection of $X(\lambda)$ to $U_q(\mathfrak{gl}(m|n))^-$ and $K(\lambda)$, respectively., Comment: 43 pages

The main purpose of this paper is to give a combinatorial realization of Kirillov-Reshetikhin (KR simply) crystals $B^{r, s}$ for type $\text{E}_n^{(1)}$ with a minuscule node $r$ and $s \ge 1$. To do this, we describe explicitly the crystal of the quantum nilpotent subalgebra associated with the translation by the negative of the $r$-th fundamental weight. Then the crystal can be extended as an affine crystal, in which a certain subcrystal characterized by the $\varepsilon_r^*$-statistic is isomorphic to $B^{r,s}$ as an affine crystal, where $\varepsilon_r^*$ is also realized precisely in terms of triple and quadruple paths., Comment: 35 pages, Abstract and Introduction are revised, rearranged Sections, revised several sentences, added several references, added Remark 5.4, added comment in Remark 6.5, typos corrected, minor corrections, v4; minor corrections, to appear in Journal of Algebra

We give a new realization of the prefundamental representations $L^\pm_{r,a}$ introduced by Hernandez and Jimbo, when the quantum loop algebra $U_q(\mathfrak{g})$ is of types $A_n^{(1)}$ and $D_n^{(1)}$, and the $r$-th fundamental weight $\varpi_r$ for types $A_n$ and $D_n$ is minuscule. We define an action of the Borel subalgebra $U_q(\mathfrak{b})$ of $U_q(\mathfrak{g})$ on the unipotent quantum coordinate ring associated to the translation by $-\varpi_r$, and show that it is isomorphic to $L^\pm_{r,a}$. We then give a combinatorial realization of $L^+_{r,a}$ in terms of the Lusztig data of the dual PBW vectors., Comment: 41 pages, introduction is revised, Lemma 3.2 is revised, the proofs of Lemma 3.8 and Corollary 4.8 are revised, added Example 3.9, Example 4.4 and Example 4.19, comments added on Remark 3.11, Remark 4.9, Remark 4.21 and Remark 4.30, typos corrected, minor corrections

We describe the embedding from the crystal of Kashiwara-Nakashima tableaux in type $D$ of an arbitrary shape into that of $\mathbf{i}$-Lusztig data associated to a family of reduced expressions $\mathbf{i}$ which are compatible with the maximal Levi subalgebra of type $A$. The embedding is described explicitly in terms of well-known combinatorics of type $A$ including the Sch\"{u}tzenberger's jeu de taquin and an analog of RSK algorithm., Comment: 35 pages, typos corrected, comments added on Remark 5.3 and Remark 5.4, to appear in Algebras and Representation Theory

We give a new formula for the branching rule from ${\rm GL}_n$ to ${\rm O}_n$ generalizing the Littlewood's restriction formula. The formula is given in terms of Littlewood-Richardson tableaux with certain flag conditions which vanish in a stable range. As an application, we give a combinatorial formula for the Lusztig $t$-weight multiplicity $K_{\mu 0}(t)$ of type $B_n$ and $D_n$ with highest weight $\mu$ and weight $0$., Comment: 50 pages, revised introduction, reference added, Corollary 4.12 added, comments added on Remark 3.22, 4.15, 4.19 and 5.8, comments added for Sections 3 and 6, to appear in Journal of Combinatorial Theory, Series A

We study the crystal of quantum nilpotent subalgebra of $U_q(D_n)$ associated to a maximal Levi subalgebra of type $A_{n-1}$. We show that it has an affine crystal structure of type $D_n^{(1)}$ isomorphic to a limit of perfect Kirillov-Reshetikhin crystal $B^{n,s}$ for $s\geq 1$, and give a new polytope realization of $B^{n,s}$. We show that an analogue of RSK correspondence for type $D$ due to Burge is an isomorphism of affine crystals and give a generalization of Greene's formula for type $D$., Comment: 33 pages, reference corrected, minor correction, Proposition 3.2 is revised with minor corrections, comments added on Remark 3.3 and Remark 4.12, typos corrected, to appear in Journal of Combinatorial Theory, Series A

In the present work, we study the entropy productions in a system controlled by double control parameters. By introducing a thermal fluctuation part, we solve the problem that the second law of the thermodynamics seems to be violated by the thermal fluctuation near equilibrium in the microscopic levels. Then we define the negative and the compensating entropy productions in the macroscopic levels.

In statistical physics, entropy is generally logarithm of probability. Therefore, if dynamics is decomposed by log, entropy production should be decomposed properly. In the present work, log-decomposition of dynamics is introduced. By which time evolution operator is logarithmically decomposed into a symmetric operator and an asymmetric factor. Path probability and path entropy production are also systematically and intuitively decomposed into symmetric and asymmetric parts. From symmetric operator, non-adiabatic entropy production is derived, whereas adiabatic entropy production is from asymmetric factor.