arXiv subjects
Publications and source records attributed to Motoki Takigiku.
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 G\"ollnitz-Gordon (mod 8), Rogers-Ramanujan (mod 5) and (partially) Nandi (mod 14).
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 $A^{(2)}_{13}$.
We give an explanation for the Pieri coefficients for the stable and dual stable Grothendieck polynomials; their non-leading terms are obtained by taking an alternating sum of meets (or joins) of their leading terms.
The dual stable Grothendieck polynomials $g_λ$ and their sums $\sum_{μ\subsetλ} g_μ$ (which represent $K$-homology classes of boundary ideal sheaves and structure sheaves of Schubert varieties in the Grassmannians) have the same product structure constants. In this paper we first explain that the ring automorphism $g_λ\mapsto\sum_{μ\subsetλ} g_μ$ on the ring of symmetric functions is described as the operator $F^\perp$, the adjoint of the multiplication $(F\cdot)$, by a "group-like" element $F=\sum_{i} h_i$ where $h_i$ is the complete symmetric function. Next we give a generalization: starting with another "group-like" elements $\sum_{i} t^i h_i$, we obtain a deformation with a parameter $t$ of the ring automorphism above, as well as identities involving stable and dual stable Grothendieck polynomials.
We give a Pieri-type formula for the sum of $K$-$k$-Schur functions $\sum_{μ\leλ} g^{(k)}_μ$ over a principal order ideal of the poset of $k$-bounded partitions under the strong Bruhat order, which sum we denote by $\widetilde{g}^{(k)}_λ$. As an application of this, we also give a $k$-rectangle factorization formula $\widetilde{g}^{(k)}_{R_t\cupλ}=\widetilde{g}^{(k)}_{R_t} \widetilde{g}^{(k)}_λ$ where $R_t=(t^{k+1-t})$, analogous to that of $k$-Schur functions $s^{(k)}_{R_t\cupλ}=s^{(k)}_{R_t}s^{(k)}_λ$.
We give some new formulas about factorizations of $K$-$k$-Schur functions $g^{(k)}_λ$, analogous to the $k$-rectangle factorization formula $s^{(k)}_{R_t\cupλ}=s^{(k)}_{R_t}s^{(k)}_λ$ of $k$-Schur functions, where $λ$ is any $k$-bounded partition and $R_t$ denotes the partition $(t^{k+1-t})$ called \textit{$k$-rectangle}. Although a formula of the same form does not hold for $K$-$k$-Schur functions, we can prove that $g^{(k)}_{R_t}$ divides $g^{(k)}_{R_t\cupλ}$, and in fact more generally that $g^{(k)}_{P}$ divides $g^{(k)}_{P\cupλ}$ for any multiple $k$-rectangles $P=R_{t_1}^{a_1}\cup\dots\cup R_{t_m}^{a_m}$ and any $k$-bounded partition $λ$. We give the factorization formula of such $g^{(k)}_{P}$ and the explicit formulas of $g^{(k)}_{P\cupλ}/g^{(k)}_{P}$ in some cases, including the case where $λ$ is a partition with a single part as the easiest example.
Subsequently to the author's preceding paper, we give full proofs of some explicit formulas about factorizations of $K$-$k$-Schur functions associated with any multiple $k$-rectangles.