arXiv subjects
Publications and source records attributed to Shunsuke Tsuchioka.
We propose Andrews-Gordon type series for certain level 2 standard modules of type $A^{(2)}_{\textrm{odd}}$, and prove the corresponding sum-product identities except for $A^{(2)}_{6n+3}$. These identities generalize the identities of Göllnitz-Gordon (mod 8), Rogers-Ramanujan (mod 5) and (partially) Nandi (mod 14).
In a series of two papers, S. Capparelli, A. Meurman, A. Primc, M. Primc (CMPP) and then M. Primc put forth three remarkable sets of conjectures, stating that the generating functions of coloured integer partition in which the parts satisfy restrictions on the multiplicities admit simple infinite product forms. While CMPP related one set of conjectures to the principally specialised characters of standard modules for the affine Lie algebra $\mathrm{C}_n^{(1)}$, finding a Lie-algebraic interpretation for the remaining two sets remained an open problem. In this paper, we use the work of Griffin, Ono and the fourth author on Rogers-Ramanujan identities for affine Lie algebras to solve this problem, relating the remaining two sets of conjectures to non-standard specialisations of standard modules for $\mathrm{A}_{2n}^{(2)}$ and $\mathrm{D}_{n+1}^{(2)}$. We also use their work to formulate conjectures for the bivariate generating function of one-parameter families of CMPP partitions in terms of Hall-Littlewood symmetric functions. We make a detailed study of several further aspects of CMPP partitions, obtaining (i) functional equations for bivariate generating functions which generalise the well-known Rogers-Selberg equations, (ii) a partial level-rank duality in the $\mathrm{A}_{2n}^{(2)}$ case, and (iii) (conjectural) identities of the Rogers-Ramanujan type for $\mathrm{D}_3^{(2)}$.
We give manifestly positive Andrews-Gordon type series for the level 3 standard modules of the affine Lie algebra of type $A^{(1)}_2$. We also give corresponding bipartition identities, which have representation theoretic interpretations via the vertex operators. Our proof is based on the Borodin product formula, the Corteel-Welsh recursion for the cylindric partitions, a $q$-version of Sister Celine's technique and a generalization of Andrews' partition ideals by finite automata due to Takigiku and the author.
We define a length function for a perfect crystal. As an application, we derive a variant of the Rogers-Ramanujan identities which involves (a $q$-analog of) the Fibonacci numbers.
Motivated by spin modular representations of the symmetric groups, we propose two generalizations of the Schur regular partitions for an odd integer $p\geq 3$. One forms a subset of the set of $p$-strict partitions, and the other forms that of strict partitions. We prove that each set has a basic $A^{(2)}_{p-1}$-crystal structure. For $p=3$, it reproves Schur's 1926 partition theorem, a mod 6 analog of Rogers-Ramanujan partition theorem (RRPT). For $p=5$, it gives a computer-free proof of a conjecture by Andrews during his 3-parameter generalization of RRPT, which was first proved by Andrews-Bessenrodt-Olsson.
We reformulate the Kanade-Russell conjecture modulo 9 via the vertex operators for the level 3 standard modules of type $D^{(3)}_{4}$. Along the same line, we arrive at three partition theorems which may be regarded as an $A^{(2)}_{4}$ analog of the conjecture. One had been proven by Andrews-van Ekeren-Heluani and we point out that the others are easily proved from their results.
We give another proof of the second Rogers-Ramanujan identity by Kashiwara crystals.
We generalize the theory of linked partition ideals due to Andrews using finite automata in formal language theory and apply it to prove three Rogers--Ramanujan type identities of modulo 14 that were posed by Nandi through vertex operator theoretic construction of the level 4 standard modules of the affine Lie algebra $A^{(2)}_{2}$.
We give Andrews-Gordon type series for the principal characters of the level 5 and 7 standard modules of the affine Lie algebra $A^{(2)}_{2}$. We also give conjectural series for some level 2 modules of $A^{(2)}_{13}$.
Stembridge characterized regular crystals associated with a simply-laced generalized Cartan matrix (GCM) in terms of local graph-theoretic quantities. We give a similar axiomatization for $B_2$ regular crystals and thus for regular crystals associated with a finite GCM except $G_2$ and an affine GCM except $A^{(1)}_{1},G^{(1)}_{2},A^{(2)}_{2},D^{(3)}_4$.
We verify the Broué-Malle-Rouquier (BMR) freeness for cyclotomic Hecke algebras associated with complex reflection groups $G_{17}$, $G_{18}$, $G_{19}$ in the Shephard-Todd classification. Together with results of Ariki, Ariki-Koike, Broué-Malle, Marin, Marin-Pfeiffer and Chavli, this settled affirmatively the BMR freeness conjecture. Our verification is inspired by Bergman's diamond lemma and requires 1GB memory and 5 days calculation on a PC.
We consider graded Cartan matrices of the symmetric groups and the Iwahori-Hecke algebras of type A, which have entries in the ring $\mathbb Z[v,v^{-1}]$. These matrices may also be interpreted as Gram matrices of the Shapovalov form on sums of weight spaces of a basic representation of an affine quantum group. We present a conjecture predicting the invariant factors of these matrices and give evidence for the conjecture by proving its implications under a localization and certain specializations of the ring $\mathbb Z[v,v^{-1}]$. This proves and generalizes a conjecture of Ando-Suzuki-Yamada on the invariants of these matrices over $\mathbb Q[v,v^{-1}]$ and also generalizes the first author's recent proof of the Külshammer-Olsson-Robinson conjecture over $\mathbb Z$.
We prove that the multiplicities of certain maximal weights of $\mathfrak{g}(A^{(1)}_{n})$-modules are counted by pattern avoidance on words. This proves and generalizes a conjecture of Misra-Rebecca. We also prove similar phenomena in types $A^{(2)}_{2n}$ and $D^{(2)}_{n+1}$. Both proofs are applications of Kashiwara's crystal theory.
We introduce a new family of superalgebras which should be considered as a super version of the Khovanov-Lauda-Rouquier algebras. Let $I$ be the set of vertices of a Dynkin diagram with parity. To this data, we associate a family of graded superalgebras, the quiver Hecke superalgebras. When there are no odd vertices, these algebras are nothing but the usual Khovanov-Lauda-Rouquier algebras. We then define another family of graded superalgebras, the quiver Hecke-Clifford superalgebras, and show that they are weakly Morita superequivalent to the quiver Hecke superalgebras. Moreover, we prove that the affine Hecke-Clifford superalgebras, as well as their degenerate version, the affine Sergeev superalgebras, are isomorphic to the quiver Hecke-Clifford superalgebras after a completion.
We give the graded Cartan determinants of the symmetric groups. Based on it, we propose a gradation of Hill's conjecture which is equivalent to Külshammer-Olsson-Robinson's conjecture on the generalized Cartan invariants of the symmetric groups.
Brundan and Kleshchev showed that some parts of the representation theory of the affine Hecke-Clifford superalgebras and its finite-dimensional "cyclotomic" quotients are controlled by the Lie theory of type $A^{(2)}_{2l}$ when the quantum parameter $q$ is a primitive $(2l+1)$-th root of unity. We show in this paper that similar theorems hold when $q$ is a primitive $4l$-th root of unity by replacing the Lie theory of type $A^{(2)}_{2l}$ with that of type $D^{(2)}_{l}$.
Let $\{B(Λ_m)|m\in\Z/e\Z\}$ be the set of level one $\mathfrak{g}(A^{(1)}_{e-1})$-crystals, and consider the realization of $B(Λ_m)$ using $e$-restricted partitions. We prove a purely Young diagrammatic criterion for an element of $B(Λ_0)^{\otimes d_1}\otimes B(Λ_m)^{\otimes d_2}$ to be in the component $B(d_1Λ_0+d_2Λ_m)$. As an application, we give a non-recursive characterization of simple modules of the Hecke algebra of type $B$. In the course of the proof, we also obtain a combinatorial description of the second type of Kashiwara's Demazure crystal in $B(Λ_m)$.
We give a modular branching rule for certain wreath products as a generalization of Kleshchev's modular branching rule for the symmetric groups. Our result contains a modular branching rule for the complex reflection groups $G(m,1,n)$ (which are often called the generalized symmetric groups) in splitting fields for $\mathbb{Z}/m\mathbb{Z}$. Especially for $m=2$ (which is the case of the Weyl groups of type $B$), we can give a modular branching rule in any field. Our proof is elementary in that it is essentially a combination of Frobenius reciprocity, Mackey theorem, Clifford's theory and Kleshchev's modular branching rule.