Quadratic relations of the intertwiners are given explicitly in two cases of chiral conformal field theory, and monomial bases of the representation spaces are constructed by using the Fourier components of the intertwiners. The two cases are the (p,p')-minimal series for the Virasoro algebra where 1<p'/p<2, and the level k integrable highest weight modules for the affine Lie algebra \hat{sl}_2.
Let \{M_{r,s}\}_{0< r < p, 0< s < p'} be the irreducible Virasoro modules in the $(p,p')$-minimal series. In our previous paper, we have constructed a monomial basis of \oplus_{r=1}^{p-1}M_{r,s} in the case of $1<p'/p<2$. By `monomials' we mean vectors of the form ϕ^{(r_L,r_{L-1})}_{-n_L}...ϕ^{(r_1,r_{0})}_{-n_1} |r_0,s >, where ϕ_{-n}^{(r',r)} are the Fourier components of the (2,1)-primary field mapping M_{r,s} to M_{r',s}, and |r_0,s > is the highest weight vector of M_{r_0,s}. In this article, for all p<p' with p>2 and s=1, we describe a subset of such monomials which conjecturally forms a…
The filtration of the Virasoro minimal series representations M^{(p,p')}_{r,s} induced by the (1,3)-primary field $ϕ_{1,3}(z)$ is studied. For 1< p'/p< 2, a conjectural basis of M^{(p,p')}_{r,s} compatible with the filtration is given by using monomial vectors in terms of the Fourier coefficients of $ϕ_{1,3}(z)$. In support of this conjecture, we give two results. First, we establish the equality of the character of the conjectural basis vectors with the character of the whole representation space. Second, for the unitary series (p'=p+1), we establish for each $m$ the equality between the char…
We construct a monomial basis of the positive part of the quantized enveloping algebra associated to a finite-dimensional simple Lie algebra. As an application we give a simple proof of the existence and uniqueness of the canonical basis of the positive part.
A unipotent triangular relationship is established between the dual standard monomial theoretic basis and canonical basis for the negative part of the quantized universal enveloping algebra of type A.
Assume that $V$ is a braided vector space with diagonal type. It is shown that a monomial belongs to Nichols braided Lie algebra $\mathfrak L(V)$ if and only if this monomial is connected. A basis of Nichols braided Lie algebra and dimension of Nichols braided Lie algebra of finite Cartan type are obtained.
The Lie conformal algebra of loop Virasoro algebra, denoted by $\mathscr{CW}$, is introduced in this paper. Explicitly, $\mathscr{CW}$ is a Lie conformal algebra with $\mathbb{C}[\partial]$-basis $\{L_i\,|\,i\in\mathbb{C}\}$ and $λ$-brackets $[L_i\, {}_λ\, L_j]=(-\partial-2λ) L_{i+j}$. Then conformal derivations of $\mathscr{CW}$ are determined. Finally, rank one conformal modules and $\mathbb{Z}$-graded free intermediate series modules over $\mathscr{CW}$ are classified.
We study a vertex operator algebra (VOA) V related to the M(3,p) Virasoro minimal series. This VOA reduces in the simplest case p=4 to the level two integrable vacuum module of $\hat{sl}_2$. On V there is an action of a commutative current a(z), which is an analog of the current e(z) of $\hat{sl}_2$. Our main concern is the subspace W generated by this action from the highest weight vector of V. Using the Fourier components of a(z), we present a monomial basis of W and a semi-infinite monomial basis of V. We also give a Gordon type formula for their characters.
All the 62 monomial elements in the canonical basis B of the quantized enveloping algebra for type $A_4$ have been determined in [2]. According to Lusztig's idea [7], the elements in the canonical basis B consist of monomials and linear combinations of monomials (for convenience, we call them polynomials). In this note, we compute all the 144 polynomial elements in one variable in the canonical basis B of the quantized enveloping algebra for type $A_4$ based on our joint note [2]. We conjecture that there are other polynomial elements in two or three variables in the canonical basis B, which i…
We investigate the affine canonical basis and the monomial basis constructed in [LXZ] in Lusztig's geometric setting. We show that the transition matrix between the two bases is upper triangular with 1's in the diagonal and coefficients in the upper diagonal entries in N[v, v^{-1}]. As a consequence, we show that part of the monomial basis elements give rise to resolutions of support varieties of the affine canonical basis elements as simple perverse sheaves.
In 1990 Beilinson, Lusztig and MacPherson provided a geometric realization of modified quantum $\mathfrak{gl}_n$ and its canonical basis. A key step of their work is a construction of a monomial basis. Recently, Du and Fu provided an algebraic construction of the canonical basis for modified quantum affine $\mathfrak{gl}_n$, which among other results used an earlier construction of monomial bases using Ringel-Hall algebra of the cyclic quiver. In this paper, we give an elementary algebraic construction of a monomial basis for affine Schur algebras and modified quantum affine $\mathfrak{gl}_n$.
On the basis of the Andrews--Bailey construction, we derive fermionic sum representations of Virasoro characters of non unitary minimal models ${\cal M}(k,kp+p-1)$ and ${\cal M}(k,kp+1)$. These expressions include certain expressions conjectured by the Stony Brook group as special cases.
Kang, Kashiwara, Kim and Oh have proved that cluster monomials lie in the dual canonical basis, under a symmetric type assumption. This involves constructing a monoidal categorification of a quantum cluster algebra using representations of KLR algebras. We use a folding technique to generalise their results to all Lie types.
In this paper we show that there is a link between the combinatorics of the canonical basis of a quantized enveloping algebra and the monomial bases of the second author arising from representations of quivers. We prove that some reparametrization functions of the canonical basis, arising from the link between Lusztig's approach to the canonical basis and the string parametrization of the canonical basis, are given on a large region of linearity by linear functions arising from these monomial basis for a quantized enveloping algebra.
We present a vertex operator algebra which is an extension of the level $k$ vertex operator algebra for the $\hat{sl}_2$ conformal field theory. We construct monomial basis of its irreducible representations.
We will introduce an $\mathbb{N}$-filtration on the negative part of a quantum group of type $A_n$, such that the associated graded algebra is a q-commutative polynomial algebra. This filtration is given in terms of the representation theory of quivers, by realizing the quantum group as the Hall algebra of a quiver. We show that the induced associated graded module of any simple finite-dimensional module (of type 1) is isomorphic to a quotient of this polynomial algebra by a monomial ideal, and we provide a monomial basis for this associated graded module. This construction can be viewed as a …
We use the Jack symmetric functions as a basis of the Fock space, and study the action of the Virasoro generators $L_n$. We calculate explicitly the matrix elements of $L_n$ with respect to the Jack-basis. A combinatorial procedure which produces these matrix elements is conjectured. As a limiting case of the formula, we obtain a Pieri-type formula which represents a product of a power sum and a Jack symmetric function as a sum of Jack symmetric functions. Also, a similar expansion was found for the case when we differentiate the Jack symmetric functions with respect to power sums. As an appli…
We study consequences of a monoidal categorification of the unipotent quantum coordinate ring $A_q(\mathfrak{n}(w))$ together with the Laurent phenomenon of cluster algebras. We show that if a simple module $S$ in the category $\mathcal C_w$ strongly commutes with all the cluster variables in a cluster $[ \mathscr C]$, then $[S]$ is a cluster monomial in $[ \mathscr C ]$. If $S$ strongly commutes with cluster variables except exactly one cluster variable $[M_k]$, then $[S]$ is either a cluster monomial in $[\mathscr C ]$ or a cluster monomial in $μ_k([ \mathscr C ])$. We give a new proof of th…
