Factorization formulas of $K$-$k$-Schur functions I

Motoki Takigiku

arXiv:1704.08643·math.CO·April 28, 2017

Factorization formulas of $K$-$k$-Schur functions I

Motoki Takigiku

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TL;DR

This paper introduces new factorization formulas for $K$-$k$-Schur functions, showing divisibility properties and explicit formulas for certain partitions, extending known results for $k$-Schur functions.

Contribution

It establishes divisibility and factorization formulas for $K$-$k$-Schur functions related to multiple $k$-rectangles, generalizing previous $k$-rectangle factorizations.

Findings

01

$g^{(k)}_{R_t}$ divides $g^{(k)}_{R_t ext{cup} ext{lambda}}$

02

Explicit formulas for $g^{(k)}_{P ext{cup} ext{lambda}}/g^{(k)}_{P}$ in special cases

03

Generalization of $k$-rectangle factorization to $K$-$k$-Schur functions

Abstract

We give some new formulas about factorizations of $K$ - $k$ -Schur functions $g_{λ}^{(k)}$ , analogous to the $k$ -rectangle factorization formula $s_{R_{t} \cup λ}^{(k)} = s_{R_{t}}^{(k)} s_{λ}^{(k)}$ 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 $g_{R_{t}}^{(k)}$ divides $g_{R_{t} \cup λ}^{(k)}$ , and in fact more generally that $g_{P}^{(k)}$ divides $g_{P \cup λ}^{(k)}$ 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 $g_{P}^{(k)}$ and the explicit formulas of $g_{P \cup λ}^{(k)} / g_{P}^{(k)}$ in some cases, including the case where $λ$ is a partition with a single…

Tables2

Table 1. Table 1 . k = 3 𝑘 3 k=3 . The table of g Q ∪ λ ( 3 ) / g Q ( 3 ) subscript superscript 𝑔 3 𝑄 𝜆 subscript superscript 𝑔 3 𝑄 {g^{(3)}_{Q\cup\lambda}}/{g^{(3)}_{Q}} for Q = R t 1 ∪ ⋯ ∪ R t n 𝑄 subscript 𝑅 subscript 𝑡 1 ⋯ subscript 𝑅 subscript 𝑡 𝑛 Q=R_{t_{1}}\cup\dots\cup R_{t_{n}} ( 1 ≤ t 1 < ⋯ < t n ≤ k 1 subscript 𝑡 1 ⋯ subscript 𝑡 𝑛 𝑘 1\leq t_{1}<\dots<t_{n}\leq k ), λ ⊂ ( 1 2 2 1 3 0 ) 𝜆 superscript 1 2 superscript 2 1 superscript 3 0 \lambda\subset(1^{2}2^{1}3^{0})


$+ \emptyset$		$+ + \emptyset$	$+$	$+ + +$
$+ \emptyset$	$+ + \emptyset$	$+ + \emptyset$	$+ + + + \emptyset$	$+ + + + + + + \emptyset$
$+ \emptyset$	$+ + \emptyset$		$+$	$+ + +$
$+ 2 \emptyset$	$+ + \emptyset$	$+ 2 + 3 \emptyset$	$+ + 2 + 2 + 2 \emptyset$	$+ 2 + 2 + + 3 + 2 + 3 + 3 \emptyset$
$+ 2 \emptyset$	$+ + \emptyset$	$+ + \emptyset$	$+ + + + \emptyset$	$+ 2 + 2 + + 2 + 2 + 2 + 2 \emptyset$
$+ 2 \emptyset$	$+ 2 + 3 \emptyset$	$+ + \emptyset$	$+ 2 + + 2 + 2 \emptyset$	$+ + 2 + 2 + 2 + 2 + 3 + 3 \emptyset$
$+ 3 \emptyset$	$+ 2 + 3 \emptyset$	$+ 2 + 3 \emptyset$	$+ 2 + 2 + 4 + 5 \emptyset$	$+ 2 + 3 + 2 + 5 + 5 + 8 + 9 \emptyset$

Table 2. Table 2 . k = 3 𝑘 3 k=3 . The table of minindex ( λ , t ) minindex 𝜆 𝑡 \mathop{\mathrm{minindex}}(\lambda,t) for | λ | ≤ 6 𝜆 6 |\lambda|\leq 6 .

$λ$	$𝔠 (λ)$	$minindex$ , $𝔠 (minindex)$
		$t = 1$		$t = 2$		$t = 3$
$\emptyset$	$\emptyset$	$\emptyset$	$\emptyset$	$\emptyset$	$\emptyset$	$\emptyset$	$\emptyset$
		$\emptyset$	$\emptyset$	$\emptyset$	$\emptyset$	$\emptyset$	$\emptyset$
				$\emptyset$	$\emptyset$	$\emptyset$	$\emptyset$
		$\emptyset$	$\emptyset$	$\emptyset$	$\emptyset$
						$\emptyset$	$\emptyset$
				$\emptyset$	$\emptyset$
		$\emptyset$	$\emptyset$
						$\emptyset$	$\emptyset$
				$\emptyset$	$\emptyset$
				$\emptyset$	$\emptyset$
		$\emptyset$	$\emptyset$
						$\emptyset$	$\emptyset$

				$\emptyset$	$\emptyset$

		$\emptyset$	$\emptyset$
						$\emptyset$	$\emptyset$

				$\emptyset$	$\emptyset$

				$\emptyset$	$\emptyset$

		$\emptyset$	$\emptyset$

Equations320

s_{R_{t} \cup λ}^{(k)} = s_{R_{t}}^{(k)} s_{λ}^{(k)} .

s_{R_{t} \cup λ}^{(k)} = s_{R_{t}}^{(k)} s_{λ}^{(k)} .

g_{R_{t} \cup (r)}^{(k)} = ⎩ ⎨ ⎧ g_{R_{t}}^{(k)} \cdot g_{(r)}^{(k)} g_{R_{t}}^{(k)} \cdot (g_{(r)}^{(k)} + g_{(r - 1)}^{(k)} + \dots + g_{\emptyset}^{(k)}) (if t < r), (if t \geq r)

g_{R_{t} \cup (r)}^{(k)} = ⎩ ⎨ ⎧ g_{R_{t}}^{(k)} \cdot g_{(r)}^{(k)} g_{R_{t}}^{(k)} \cdot (g_{(r)}^{(k)} + g_{(r - 1)}^{(k)} + \dots + g_{\emptyset}^{(k)}) (if t < r), (if t \geq r)

g_{R_{t} \cup R_{t}}^{(k)} = g_{R_{t}}^{(k)} λ \subset R_{t} \sum g_{λ}^{(k)} .

g_{R_{t} \cup R_{t}}^{(k)} = g_{R_{t}}^{(k)} λ \subset R_{t} \sum g_{λ}^{(k)} .

g_{R_{t_{1}}^{a_{1}} \cup \dots \cup R_{t_{m}}^{a_{m}}}^{(k)} = g_{R_{t_{1}}}^{(k)} λ^{(1)} \subset R_{t_{1}} \sum g_{λ^{(1)}}^{(k)}^{a_{1} - 1} \dots g_{R_{t_{m}}}^{(k)} λ^{(m)} \subset R_{t_{m}} \sum g_{λ^{(m)}}^{(k)}^{a_{m} - 1} .

g_{R_{t_{1}}^{a_{1}} \cup \dots \cup R_{t_{m}}^{a_{m}}}^{(k)} = g_{R_{t_{1}}}^{(k)} λ^{(1)} \subset R_{t_{1}} \sum g_{λ^{(1)}}^{(k)}^{a_{1} - 1} \dots g_{R_{t_{m}}}^{(k)} λ^{(m)} \subset R_{t_{m}} \sum g_{λ^{(m)}}^{(k)}^{a_{m} - 1} .

\overset{ˉ}{λ}

\overset{ˉ}{λ}

(4, 2)

(4, 2)

λ^{(n)}

λ^{(n)}

P_{k} \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces

P_{k} \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces

s : \tilde{S}_{k + 1} / S_{k + 1} ⟶ C_{k + 1}; w \mapsto w \cdot \emptyset .

s : \tilde{S}_{k + 1} / S_{k + 1} ⟶ C_{k + 1}; w \mapsto w \cdot \emptyset .

C_{k + 1} ⟶ P_{k} ⟶ \tilde{S}_{k + 1} / S_{k + 1}; κ \mapsto p (κ) \mapsto w_{p (κ)},

C_{k + 1} ⟶ P_{k} ⟶ \tilde{S}_{k + 1} / S_{k + 1}; κ \mapsto p (κ) \mapsto w_{p (κ)},

w ≺ \cdot v : ⟺ \exists i such that s_{i} w = v, l (w) + 1 = l (v) .

w ≺ \cdot v : ⟺ \exists i such that s_{i} w = v, l (w) + 1 = l (v) .

P_{k}

P_{k}

C_{k + 1}

h_{r} s_{λ} = μ / λ :horizontal r -strip \sum s_{μ} .

h_{r} s_{λ} = μ / λ :horizontal r -strip \sum s_{μ} .

s_{\emptyset}^{(k)}

s_{\emptyset}^{(k)}

h_{r} s_{λ}^{(k)}

h_{r} \cdot g_{λ}^{(k)} = (μ, ρ) \sum (- 1)^{∣ λ ∣ + r - ∣ μ ∣} g_{μ}^{(k)},

h_{r} \cdot g_{λ}^{(k)} = (μ, ρ) \sum (- 1)^{∣ λ ∣ + r - ∣ μ ∣} g_{μ}^{(k)},

h_{r} \cdot g_{λ}^{(k)} = s = 0 \sum r (- 1)^{r - s} μ c (μ) / c (λ) : weak s -strip \sum (r - s r _{c (μ) c (λ)}) g_{μ}^{(k)} .

h_{r} \cdot g_{λ}^{(k)} = s = 0 \sum r (- 1)^{r - s} μ c (μ) / c (λ) : weak s -strip \sum (r - s r _{c (μ) c (λ)}) g_{μ}^{(k)} .

λ \cup P ⪯ μ ⟺ \exists ν \in P_{k}, {μ = ν \cup P, λ ⪯ ν .

λ \cup P ⪯ μ ⟺ \exists ν \in P_{k}, {μ = ν \cup P, λ ⪯ ν .

c (ν) / c (λ) is a weak strip ⟺ c (ν \cup P) / c (λ \cup P) is a weak strip

c (ν) / c (λ) is a weak strip ⟺ c (ν \cup P) / c (λ \cup P) is a weak strip

c (μ) / c (λ \cup P) is a weak strip ⟺ \exists ν \in P_{k}, {μ = ν \cup P, c (ν) / c (λ) is a weak strip .

c (μ) / c (λ \cup P) is a weak strip ⟺ \exists ν \in P_{k}, {μ = ν \cup P, c (ν) / c (λ) is a weak strip .

c (\tilde{λ})_{i} = {c (λ)_{i} + t c (λ)_{i - (k + 1 - t)} (if i \leq r + (k + 1 - t)), (if i \geq (r + 1) + (k + 1 - t)) .

c (\tilde{λ})_{i} = {c (λ)_{i} + t c (λ)_{i - (k + 1 - t)} (if i \leq r + (k + 1 - t)), (if i \geq (r + 1) + (k + 1 - t)) .

c (\tilde{λ})_{i}

c (\tilde{λ})_{i}

= c (\tilde{λ})_{i + k + 1 - t} + t (since \tilde{λ}_{i} = t)

= c (λ)_{i} + t . (by the latter case)

c (\tilde{λ})_{i}

c (\tilde{λ})_{i}

= c (\tilde{λ})_{\leq r + k + 1 - t i + k + 1 - λ_{i}} + λ_{i}

= c (λ)_{i + k + 1 - λ_{i}} + t + λ_{i}

= c (λ)_{i} + t .

P

P

α_{P} (u)

g_{P \cup \hat{λ}}^{(k)} \cdot g_{(λ_{l})}^{(k)} = g_{P \cup λ}^{(k)} + μ \sum a_{λ μ} g_{P \cup μ}^{(k)}

g_{P \cup \hat{λ}}^{(k)} \cdot g_{(λ_{l})}^{(k)} = g_{P \cup λ}^{(k)} + μ \sum a_{λ μ} g_{P \cup μ}^{(k)}

g_{P \cup λ}^{(k)} = g_{P}^{(k)} (g_{λ}^{(k)} + μ \sum a_{λ μ} g_{μ}^{(k)}),

g_{P \cup λ}^{(k)} = g_{P}^{(k)} (g_{λ}^{(k)} + μ \sum a_{λ μ} g_{μ}^{(k)}),

h_{r} \cdot g_{λ}^{(k)} = s = 0 \sum r (- 1)^{r - s} ν c (ν) / c (λ) :weak s -strip \sum (r - s r _{c (ν) c (λ)}) g_{ν}^{(k)},

h_{r} \cdot g_{λ}^{(k)} = s = 0 \sum r (- 1)^{r - s} ν c (ν) / c (λ) :weak s -strip \sum (r - s r _{c (ν) c (λ)}) g_{ν}^{(k)},

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TopicsAdvanced Combinatorial Mathematics · Algebraic structures and combinatorial models · Advanced Mathematical Identities

Full text

Factorization formulas of $K$ - $k$ -Schur functions I

Motoki TAKIGIKU

Graduate School of Mathematical Sciences, the University of Tokyo, Japan

Abstract.

We give some new formulas about factorizations of $K$ - $k$ -Schur functions $g^{(k)}_{\lambda}$ , analogous to the $k$ -rectangle factorization formula $s^{(k)}_{R_{t}\cup\lambda}=s^{(k)}_{R_{t}}s^{(k)}_{\lambda}$ of $k$ -Schur functions, where $\lambda$ is any $k$ -bounded partition and $R_{t}$ denotes the partition $(t^{k+1-t})$ called $k$ -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 $g^{(k)}_{R_{t}\cup\lambda}$ , and in fact more generally that $g^{(k)}_{P}$ divides $g^{(k)}_{P\cup\lambda}$ for any multiple $k$ -rectangles $P=R_{t_{1}}^{a_{1}}\cup\dots\cup R_{t_{m}}^{a_{m}}$ and any $k$ -bounded partition $\lambda$ . We give the factorization formula of such $g^{(k)}_{P}$ and the explicit formulas of $g^{(k)}_{P\cup\lambda}/g^{(k)}_{P}$ in some cases, including the case where $\lambda$ is a partition with a single part as the easiest example.

[email protected]

1 Introduction
2 Preliminaries
2.1 Partitions
2.2 Bounded partitions, cores, affine Grassmannian elements, and $k$ -rectangles $R_{t}$
2.3 Weak order and weak strips
2.4 Symmetric functions
2.5 $k$ -Schur functions
2.6 $K$ - $k$ *-Schur functions * $g^{(k)}_{\lambda}$
2.7 Some properties of bounded partitions and cores
3 Possibility of factoring out $g^{(k)}_{R_{t_{1}}^{a_{1}}\cup\dots\cup R_{t_{m}}^{a_{m}}}$ and some other general results
4 *A factorization of * $g^{(k)}_{R_{t_{1}}^{a_{1}}\cup\dots\cup R_{t_{m}}^{a_{m}}\cup(r)}$
5 *A factorization of $g^{(k)}_{R_{t_{1}}^{a_{1}}\cup\dots\cup R_{t_{m}}^{a_{m}}\cup\lambda}$ with small $\lambda$ and splitting $g^{(k)}_{R_{t_{1}}^{a_{1}}\cup\dots\cup R_{t_{m}}^{a_{m}}}$ into * $g^{(k)}_{R_{t_{1}}^{a_{1}}}\dots g^{(k)}_{R_{t_{m}}^{a_{m}}}$
5.1 Statements
5.2 Proofs
5.3 Steps (A-1) and (B-1)
5.4 Steps (A-2) and (B-2)
5.5 Step (C)
6 Discussions
A Examples
B *Proof of Proposition *3
C *Explicit description of * $A_{\mu,{\bar{\lambda}},q}$

1. Introduction

Let $k$ be a positive integer. $K$ - $k$ -Schur functions $g^{(k)}_{\lambda}$ are inhomogeneous symmetric functions parametrized by $k$ -bounded partitions $\lambda$ , namely by the weakly decreasing strictly positive integer sequences $\lambda=(\lambda_{1},\dots,\lambda_{l})$ , $l\in\mathbb{Z}_{\geq 0}$ , whose terms are all bounded by $k$ . They are $K$ -theoretic analogues of another family of symmetric functions called $k$ -Schur functions, which are homogeneous and also parametrized by $k$ -bounded partitions.

Historically, $k$ -Schur functions were first introduced by Lascoux, Lapointe and Morse [MR1950481], and subsequent studies led to several (conjectually equivalent) characterizations of $s^{(k)}_{\lambda}$ such as the Pieri-like formula due to Lapointe and Morse [MR2331242], and Lam proved that $k$ -Schur functions correspond to the Schubert basis of homology of the affine Grassmannian [Lam08]. Moreover it was shown by Lam and Shimozono that $k$ -Schur functions play a central role in the explicit description of the Peterson isomorphism between quantum cohomology of the Grassmannian and homology of the affine Grassmannian up to suitable localizations [MR2923177].

These developments have analogues in $K$ -theory. Lam, Schilling and Shimozono [MR2660675] characterized the $K$ -theoretic $k$ -Schur functions as the Schubert basis of the $K$ -homology of the affine Grassmannian, and Morse [Morse12] investigated them from a conbinatorial viewpoint, giving various properties including the Pieri-like formulas using affine set-valued strips (the form using cyclically decreasing words are also given in [MR2660675]).

In this paper we start from this combinatorial characterization (see Definition 7) and show certain new factorization formulas of $K$ - $k$ -Schur functions.

Among the $k$ -bounded partitions, those of the form $(t^{k+1-t})=(\underbrace{t,\dots,t}_{k+1-t})=:R_{t}$ , $1\leq t\leq k$ , called $k$ -rectangle, play a special role. In particular, if a $k$ -bounded partition has the form $R_{t}\cup\lambda$ , where the symbol $\cup$ denotes the operation of concatenating the two sequences and reordering the terms in the weakly decreasing order, then the corresponding $k$ -Schur function has the following factorization property [MR2331242, Theorem 40]:

[TABLE]

Note that, under the bijection between the set of $k$ -bounded partitions and the set of affine Grassmannian elements in the affine symmetric group, the correspondent of the $k$ -rectangle $R_{i}$ is congruent, in the extended affine Weyl group, to the translation $t_{-\varpi_{i}^{\vee}}$ by the negative of a fundamental coweight, modulo left multiplication by the length zero elements.

It is suggested in [MR2660675, Remark 7.4] that the $K$ - $k$ -Schur functions should also possess similar properties, including the divisibility of $g^{(k)}_{R_{t}\cup\lambda}$ by $g^{(k)}_{R_{t}}$ . The present work is an attempt to materialize this suggestion.

We do show in Proposition 14 that $g^{(k)}_{R_{t}}$ divides $g^{(k)}_{R_{t}\cup\lambda}$ in the ring $\Lambda^{(k)}=\mathbb{Z}[h_{1},\dots,h_{k}]$ , where $h_{i}$ denotes the complete homogeneous symmetric functions of degree $i$ , of which the $K$ - $k$ -Schur functions form a basis. However, unlike the case of $k$ -Schur functions, the quotient $g^{(k)}_{R_{t}\cup\lambda}/g^{(k)}_{R_{t}}$ is not a single term $g^{(k)}_{\lambda}$ but, in general, a linear combination of $K$ - $k$ -Schur functions with leading term $g^{(k)}_{\lambda}$ , namely in which $g^{(k)}_{\lambda}$ is the only highest degree term. Even the simplest case where $\lambda$ consists of a single part $(r)$ , $1\leq r\leq k$ , displays this phenomenon: we show in Theorem 23 that

[TABLE]

(actually we have $g^{(k)}_{(s)}=h_{s}$ for $1\leq s\leq k$ , and $g^{(k)}_{\varnothing}=h_{0}=1$ ). So we may ask:

Question 1. *Which $g^{(k)}_{\mu}$ , besides $g^{(k)}_{\lambda}$ , appear in the quotient $g^{(k)}_{R_{t}\cup\lambda}/g^{(k)}_{R_{t}}$ ? With what cofficients? *

A $k$ -bounded partition can always be written in the form $R_{t_{1}}\cup\dots\cup R_{t_{m}}\cup\lambda$ with $\lambda$ not having so many repetitions of any part as to form a $k$ -rectangle. In such an expression we temporarily call $\lambda$ the remainder, although this term is only used in the Introduction. Proceeding in the direction of Question 1, one ultimate goal may be to give a factorization formula in terms of the $k$ -rectangles and the remainder. In the case of $k$ -Schur functions, the straightforward factorization in (1) above leads to the formula $s^{(k)}_{R_{t_{1}}\cup\dots\cup R_{t_{m}}\cup\lambda}=s^{(k)}_{R_{t_{1}}}\dots s^{(k)}_{R_{t_{m}}}s^{(k)}_{\lambda}$ . On the contrary, with $K$ - $k$ -Schur functions, the simplest case having a multiple $k$ -rectangle, to be shown in the author’s following paper [Takigiku], gives

[TABLE]

Hence we cannot expect $g^{(k)}_{R_{t}\cup R_{t}}$ to be divisible by $g^{(k)}_{R_{t}}$ twice. Instead, upon organizing the part consisting of $k$ -rectangles in the form $R_{t_{1}}^{a_{1}}\cup\dots\cup R_{t_{m}}^{a_{m}}$ with $t_{1}<\dots<t_{m}$ and $a_{i}\geq 1$ ( $1\leq i\leq m$ ), with $R_{t}^{a}=\underbrace{R_{t}\cup\dots\cup R_{t}}_{a}$ , actually we show in Proposition 14 that

$g^{(k)}_{R_{t_{1}}^{a_{1}}\cup\dots\cup R_{t_{m}}^{a_{m}}\cup\lambda}$ is divisible by $g^{(k)}_{R_{t_{1}}^{a_{1}}\cup\dots\cup R_{t_{m}}^{a_{m}}}$ ,

which actually holds whether or not $\lambda$ is the remainder. Then we can subdivide our goal as follows:

Question $1^{\prime}$ . *Which $g^{(k)}_{\mu}$ , besides $g^{(k)}_{\lambda}$ , appear in the quotient $g^{(k)}_{P\cup\lambda}/g^{(k)}_{P}$ where $P=R_{t_{1}}^{a_{1}}\cup\dots\cup R_{t_{m}}^{a_{m}}$ , and with what coefficients? *

Question $2$ . *How can $g^{(k)}_{R_{t_{1}}^{a_{1}}\cup\dots\cup R_{t_{m}}^{a_{m}}}$ be factorized? *

In this (and author’s following) paper, we give a reasonably complete answer to Question 2, and partial answers to Question $1^{\prime}$ . For Question 2, we first show in Theorem 31 that multiple $k$ -rectangles of different sizes entirely split, namely that we have $g^{(k)}_{R_{t_{1}}^{a_{1}}\cup\dots\cup R_{t_{m}}^{a_{m}}}=g^{(k)}_{R_{t_{1}}^{a_{1}}}\dots g^{(k)}_{R_{t_{m}}^{a_{m}}}$ . Then in the following paper [Takigiku], we show that for each $1\leq t\leq k$ and $a>1$ , we have a nice factorization $\displaystyle g^{(k)}_{R_{t}^{a}}=g^{(k)}_{R_{t}}\left(\sum_{\lambda\subset R_{t}}g^{(k)}_{\lambda}\right)^{a-1}$ , generalizing the formula (3). Thus, we have

[TABLE]

For Question $1^{\prime}$ , unfortunately we cannot give a complete answer yet. Still we obtain some nice explicit formulas, including the case (2).

We first show an auxiliary result that, being given $P=R_{t_{1}}^{a_{1}}\cup\dots\cup R_{t_{m}}^{a_{m}}$ and putting $Q=R_{t_{1}}\cup\dots\cup R_{t_{m}}$ without multiplicities, we have $g^{(k)}_{P\cup\lambda}/g^{(k)}_{P}=g^{(k)}_{Q\cup\lambda}/g^{(k)}_{Q}$ for any $\lambda$ . Thus we can reduce Question $1^{\prime}$ to the case where the $k$ -rectangles are of all different sizes.

We then derive explicit formulas in some limited cases where, writing $\lambda=(\lambda_{1},\dots,\lambda_{l})$ and $\bar{\lambda}=(\lambda_{1},\dots,\lambda_{l-1})$ , the parts of $\lambda$ except for $\lambda_{l}$ are all larger than the widths of the $k$ -rectangles, and $\bar{\lambda}$ is contained in a $k$ -rectangle.

An easiest case is where $\lambda=(r)$ consists of a single part, which generalizes the case (2). Namely we show that if $P=R_{t_{1}}^{a_{1}}\cup\dots\cup R_{t_{m}}^{a_{m}}$ with $t_{1}<\dots<t_{m}$ and $a_{1},\dots,a_{m}>0$ and $0\leq r\leq k$ , then $g^{(k)}_{P\cup(r)}$ decomposes as $g^{(k)}_{P}\cdot\sum_{s=0}^{r}\binom{\alpha_{P}(r)+s-1}{s}g^{(k)}_{(r-s)}$ , where $\alpha_{P}(u)=\#\{i\mid 1\leq i\leq m,\ t_{i}\geq u\}$ . Considering the case $m=1$ and $a_{1}=1$ , we obtain the formula (2).

Generalizing this case, we show in Theorem 30 that if $P$ and $\alpha_{P}(u)$ are the same as above and $\lambda=(\lambda_{1},\dots,\lambda_{l})$ satisfies $\lambda_{l-1}>t_{m}$ and $\bar{\lambda}=(\lambda_{1},\dots,\lambda_{l-1})$ is contained in a $k$ -rectangle, then $g^{(k)}_{P\cup\lambda}$ decomposes as $g^{(k)}_{P}\cdot\sum_{s=0}^{\lambda_{l}}\binom{\alpha_{P}(\lambda_{l})+s-1}{s}g^{(k)}_{\bar{\lambda}\cup(\lambda_{l}-s)}$ . In particular, if $t_{n}<\lambda_{l}$ , the summation on the right-hand side consists of a single term $g^{(k)}_{\lambda}$ .

[TABLE]

It is worth noting that, in all cases we have seen, $g^{(k)}_{P\cup\lambda}\big{/}g^{(k)}_{P}$ is a linear combination of $K$ - $k$ -Schur functions with positive coefficients. Moreover, if $P=R_{t}$ , it seems that each coefficient is [math] or $1$ and the set of $\mu$ such that the coefficient of $g^{(k)}_{\mu}$ in $g^{(k)}_{P\cup\lambda}\big{/}g^{(k)}_{P}$ is $1$ is an interval (with respect to the strong order. See Conjecture 33). Anyway, it should be interesting to study the geometric meaning of these results and conjectures.

This paper is organized as follows. In Section 2, we review some basic notations and facts about combinatorial backgrounds of $K$ - $k$ -Schur functions. In Section 3, we show some auxiliary results which provide a basis for our work. In Section 4, we give explicit factorization formulas in an easiest case where the remainder consists of a single part. In Section 5, we generalize the result of the previous section and give a “straightforward factorization” formula for a multiple $k$ -rectangles of different sizes. In Section 6, we state some observations and conjectures.

**Acknowledgement. ** The author would like to express his gratitude to T. Ikeda for suggesting the problem to the author and helping him with many fruitful discussions. He is grateful to H. Hosaka and I. Terada for many valuable comments and pointing out mistakes and typos in the draft version of this paper. He is also grateful to the committee of the 29th international conference on Formal Power Series and Algebraic Combinatorics (FPSAC) for many valuable comments for the extended abstract version of this paper. This work was supported by the Program for Leading Graduate Schools, MEXT, Japan. The contents of this paper is the first half of the author’s master-thesis [MasterThesis].

2. Preliminaries

In this section we review some requisite combinatorial backgrounds. For detailed definitions, see for instance [MR3379711, Chapter 2] or [MR1354144, Chapter I].

2.1. Partitions

Let $\mathcal{P}$ denote the set of partitions. A partition $\lambda=(\lambda_{1}\geq\lambda_{2}\geq\dots)\in\mathcal{P}$ is identified with its Young diagram (or shape), for which we use the French notation here. quadrant111We use the French notation of Cartesian plane so that there are $\lambda_{i}$ boxes arranged in left justified way in the $i$ -th row from the bottom.

[TABLE]

We denote the size of a partition $\lambda$ by $|\lambda|$ , the length by $l(\lambda)$ , and the conjugate by $\lambda^{\prime}$ . For partitions $\lambda$ , $\mu$ we say $\lambda\subset\mu$ if $\lambda_{i}\leq\mu_{i}$ for all $i$ . The dominance order $\unlhd$ on $\mathcal{P}$ is defined by saying that $\lambda\unlhd\mu$ if $|\lambda|=|\mu|$ and $\sum_{i=1}^{r}\lambda_{i}\leq\sum_{i=1}^{r}\mu_{i}$ for all $r\geq 1$ . Sometimes we abbreviate horizontal strip (resp. vertical strip) (of size $r$ ) to ( $r$ -)h.s. (resp. ( $r$ -)v.s.). For a partition $\lambda$ and a cell $c=(i,j)$ in $\lambda$ , we denote the hook length of $c$ in $\lambda$ by $\mathrm{hook}_{c}(\lambda)=\lambda_{i}+\lambda^{\prime}_{j}-i-j+1$ .

For a partition $\lambda$ , a removable corner of $\lambda$ (or $\lambda$ -removable corner) is a cell $(i,j)\in\lambda$ with $(i,j+1),(i+1,j)\notin\lambda$ . $(i,j)\in(\mathbb{Z}_{>0})^{2}\setminus\lambda$ is said to be an addable corner of $\lambda$ (or $\lambda$ -addable corner) if $(i,j-1),(i-1,j)\in\lambda$ with the understanding that $(0,j),(j,0)\in\lambda$ . In order to avoid making equations too wide, we may denote removable corner (resp. addable corner) briefly by rem. cor. (resp. add. cor.). A cell $(i,j)\in\lambda$ is called extremal if $(i+1,j+1)\notin\lambda$ .

For partitions $\lambda=(\lambda_{1},\ldots,\lambda_{l(\lambda)})$ , $\mu=(\mu_{1},\ldots,\mu_{l(\mu)})$ , we write $\lambda\oplus\mu=(\lambda_{1}+\mu_{1},\lambda_{2}+\mu_{1},\ldots,\lambda_{l(\lambda)}+\mu_{1},\mu_{1},\ldots,\mu_{l(\mu)})$ . For partitions $\lambda^{(1)},\ldots,\lambda^{(n)}$ , we define $\lambda^{(1)}\oplus\cdots\oplus\lambda^{(n)}=(\lambda^{(1)}\oplus\cdots\oplus\lambda^{(n-1)})\oplus\lambda^{(n)}$ , recursively.

[TABLE]

the shape of $\lambda^{(1)}\oplus\dots\oplus\lambda^{(n)}$

2.2. Bounded partitions, cores, affine Grassmannian elements, and $k$ -rectangles $R_{t}$

A partition $\lambda$ is called $k$ -bounded if $\lambda_{1}\leq k$ . Let $\mathcal{P}_{k}$ be the set of all $k$ -bounded partitions. An $r$ -core (or simply a core if no confusion can arise) is a partition none of whose cells have a hook length equal to $r$ . We denote by $\mathcal{C}_{r}$ the set of all $r$ -core partitions. When we consider a partition as a core, the notion of size differs from the usual one: the length (or size) of an $r$ -core $\kappa$ is the number of cells in $\kappa$ whose hook length is smaller than $r$ , and denoted by $|\kappa|_{r}$ .

The affine symmetric group $\tilde{S}_{k+1}$ is given by generators $\{s_{0},s_{1},\dots,s_{k}\}$ and relations $s_{i}^{2}=1$ , $s_{i}s_{i+1}s_{i}=s_{i+1}s_{i}s_{i+1}$ , $s_{i}s_{j}=s_{j}s_{i}$ for $i-j\not\equiv 0,1,k\mod(k+1)$ , with all indices are considered mod $(k+1)$ . Note that the symmetric group $S_{k+1}$ generated by $\{s_{1},\dots,s_{k}\}$ is a subgroup of $\tilde{S}_{k+1}$ . We identify the left cosets of $\tilde{S}_{k+1}/S_{k+1}$ with their minimal length representatives, which we call affine Grassmannian elements. Namely, the set of affine Grassmannian elements is $\{w\in\tilde{S}_{k+1}\mid l(ws_{i})>l(w)\ (\forall i\neq 0)\}$ .

Hereafter we fix a positive integer $k$ .

For a cell $c=(i,j)$ , the content of $c$ is $j-i$ and the residue of $c$ is $\mathrm{res}(c)=j-i\mod(k+1)\in\mathbb{Z}/(k+1)$ . For a set $X$ of cells, we write $\mathrm{Res}(X)=\{\,\mathrm{res}(c)\mid c\in X\,\}$ . We will write a $\lambda$ -removable corner of residue $i$ simply a $\lambda$ -removable $i$ -corner. For simplicity of notation, we may use an integer to represent a residue, omitting “mod $(k+1)$ ”.

We denote by $R_{t}$ the partition $(t^{k+1-t})=(t,t,\dots,t)\in\mathcal{P}_{k}$ for $1\leq t\leq k$ , which is called a $k$ -rectangle. Naturally a $k$ -rectangle is a $(k+1)$ -core.

Now we recall the bijection between the $k$ -bounded partitions in $\mathcal{P}_{k}$ , the $(k+1)$ -cores in $\mathcal{C}_{k+1}$ , and the affine Grassmannian elements in $\tilde{S}_{k+1}/S_{k+1}$ :

[TABLE]

The maps $\mathfrak{p}$ and $\mathfrak{c}$ :**

The map $\mathfrak{p}\colon\mathcal{C}_{k+1}\longrightarrow\mathcal{P}_{k};\kappa\mapsto\lambda$ is defined by $\lambda_{i}=\#\{j\mid(i,j)\in\kappa,\ \mathrm{hook}_{(i,j)}(\kappa)\leq k\}$ .

The map $\mathfrak{c}\colon\mathcal{P}_{k}\longrightarrow\mathcal{C}_{k+1};\lambda\mapsto\kappa$ is defined by the following procedure: given a $k$ -bounded partition $\lambda$ then work from the smallest part to the largest. For each row, calculate the hook lengths of all its cells. If there is a cell with hook length greater than $k$ , slide this row to the right until all its cells have hook length not greater than $k$ . In the end this process produces a skew shape $\mu/\nu$ , where in fact $\mu$ is a $(k+1)$ -core. Then let $\kappa$ be this $\mu$ .

Then in fact $\mathfrak{p}$ and $\mathfrak{c}$ are bijective and $\mathfrak{p}=\mathfrak{c}^{-1}$ . See [MR2167475, Theorem 7] for the proof. The next lemma gives a more explicit description for $\mathfrak{c}$ , which follows from the argument given just before [MR3379711, Example 1.23]:

Lemma 1.

For $\lambda\in\mathcal{P}_{k}$ and $j\geq 1$ , $\mathfrak{c}(\lambda)_{j}=\mathfrak{c}(\lambda)_{j+k+1-\lambda_{j}}+\lambda_{j}$ .

Note that if $\lambda$ is contained in a $k$ -rectangle then $\lambda\in\mathcal{P}_{k}$ and $\lambda\in\mathcal{C}_{k+1}$ , and besides $\mathfrak{p}(\lambda)=\lambda=\mathfrak{c}(\lambda)$ .

The map $\mathfrak{s}$ and the inverse:* For $\kappa\in\mathcal{C}_{k+1}$ and $i=0,1,\dots,k$ , we define $s_{i}\cdot\kappa$ as follows:*

•

if there is a $\kappa$ -addable $i$ -corner, then let $s_{i}\cdot\kappa$ be $\kappa$ with all $\kappa$ -addable $i$ -corners added,

•

if there is a $\kappa$ -removable $i$ -corner, then let $s_{i}\cdot\kappa$ be $\kappa$ with all $\kappa$ -removable $i$ -corners removed,

•

otherwise, let $s_{i}\cdot\kappa$ be $\kappa$ .

In fact first and second case never occur simultaneously and $s_{i}\cdot\kappa\in\mathcal{C}_{k+1}$ and then we have a well-defined $\tilde{S}_{k+1}$ -action on $\mathcal{C}_{k+1}$ and it induces a bijection

[TABLE]

The inverse map is given by

[TABLE]

where $w_{\lambda}$ is the affine permutation $s_{i_{1}}s_{i_{2}}\dots s_{i_{l}}$ , where $(i_{1},i_{2},\dots,i_{l})$ is the sequence obtained by reading the residues of the cells in $\lambda$ , from the shortest row to the largest, and within each row from right to left. See [MR2167475, Corollary 48] for the proof.

2.3. Weak order and weak strips

In this subsection we review the weak order on $\mathcal{P}_{k}\simeq\mathcal{C}_{k+1}\simeq\tilde{S}_{k+1}/S_{k+1}$ .

For a $k$ -bounded partition $\lambda$ , its $k$ -conjugate ${\lambda}^{\omega_{k}}$ is also a $k$ -bounded partition given by ${\lambda}^{\omega_{k}}=\mathfrak{p}(\mathfrak{c}(\lambda)^{\prime})$ .

Definition-Proposition 2.

The weak order $\prec$ on $\tilde{S}_{k+1}/S_{k+1}$ is defined by the following covering relation:

[TABLE]

It is transferred to $\mathcal{P}_{k}$ and $\mathcal{C}_{k+1}$ by the bijection described above as follows:

[TABLE]

Proof.

$(\ref{wo_Sn})\iff(\ref{wo_Cn})$ : see [MR1851953]. $(\ref{wo_Pk})\iff(\ref{wo_Cn})$ : see [MR2167475, Corollary 25]. ∎

Definition-Proposition 3.

For $(k+1)$ -cores $\tau\subset\kappa\in\mathcal{C}_{k+1}$ , $\kappa/\tau$ is called a weak strip of size $r$ (or a weak $r$ -strip) if the following equivalent conditions hold:

(1)

$\kappa/\tau$ * is horizontal strip and $\tau\prec\cdot\hskip 2.84526pt\exists\tau^{(1)}\prec\cdot\hskip 2.84526pt\dots\prec\cdot\hskip 2.84526pt\exists\tau^{(r)}=\kappa$ .* 2. (2)

$\kappa/\tau$ * is horizontal strip and $|\kappa|_{k+1}=|\tau|_{k+1}+r$ and $\#\mathrm{Res}(\kappa/\tau)=r$ .* 3. (3)

$\mathfrak{p}(\kappa)/\mathfrak{p}(\tau)$ * is a horizontal strip and $\mathfrak{p}(\kappa^{\prime})/\mathfrak{p}(\tau^{\prime})$ is a vertical strip and $|\kappa|_{k+1}=|\tau|_{k+1}+r$ .* 4. (4)

$\kappa=s_{i_{1}}\dots s_{i_{r}}\tau$ * for some cyclically decreasing element $s_{i_{1}}\dots s_{i_{r}}$ .*

(Here, an affine permutation $w=s_{i_{1}}\dots s_{i_{r}}$ (a reduced expression) is called cyclically decreasing if $i_{1},\dots,i_{r}$ are distinct and $j$ never precedes $j+1$ (taken modulo $k+1$ ) in the sequence $i_{1}i_{2}\dots i_{r}$ . This definition is in fact independent of which reduced expression we choose. )

Proof.

$(1)\Longrightarrow(3)$ : see [MR2167475, Theorem 58]. $(3)\Longrightarrow(1)$ : see [MR2167475, Proposition 54, Theorem 56]. $(3)\Longrightarrow(2)$ : see [MR2167475, Theorem 56].

$(4)\Longrightarrow(1)$ , $(1)\Longrightarrow(4)$ : see Appendix B.

$(2)\Longrightarrow(1)$ : omitted since (2) is not used in this paper. ∎

2.4. Symmetric functions

Let $\Lambda=\mathbb{Z}[h_{1},h_{2},\dots]$ be the ring of symmetric functions, generated by the complete symmetric functions $h_{r}=\sum_{i_{1}\leq i_{2}\leq\dots\leq i_{r}}x_{i_{1}}\dots x_{i_{r}}$ . For a partition $\lambda$ we set $h_{\lambda}=h_{\lambda_{1}}h_{\lambda_{2}}\dots h_{\lambda_{l(\lambda)}}$ . Then $\{h_{\lambda}\}_{\lambda\in\mathcal{P}}$ forms a $\mathbb{Z}$ -basis of $\Lambda$ . The Schur functions $\{s_{\lambda}\}_{\lambda\in\mathcal{P}}$ are the family of symmetric functions satisfying the Pieri rule:

[TABLE]

Note that $h_{r}s_{\lambda}=s_{\lambda\cup(r)}+\sum_{\mu\rhd\lambda\cup(r)}a_{\mu}s_{\mu}$ for some $a_{\mu}$ . Using this repeatedly, we can write $h_{\lambda}=s_{\lambda}+\sum_{\lambda\lhd\mu}K_{\mu\lambda}s_{\mu}$ for some coefficients $K_{\mu\lambda}$ . Thus Schur functions $\{s_{\lambda}\}_{\lambda\in\mathcal{P}}$ form a basis of $\Lambda$ since the transformation matrix between $\{s_{\lambda}\}_{\lambda}$ and $\{h_{\mu}\}_{\mu}$ is unitriangular.

2.5. $k$ -Schur functions

We recall a characterization of $k$ -Schur functions given in [MR2331242], since it is a model for and has a relationship with $K$ - $k$ -Schur functions.

Definition 4 ( $k$ -Schur function via “weak Pieri rule”).

$k$ -Schur functions $\{s^{(k)}_{\lambda}\}_{\lambda\in\mathcal{P}_{k}}$ are the family of symmetric functions such that

[TABLE]

summed over $\mu\in\mathcal{P}_{k}$ such that $\mathfrak{c}(\mu)/\mathfrak{c}(\lambda)$ is a weak strip of size $r$ .

According to the fact that if $\mathfrak{c}(\nu)/\mathfrak{c}(\eta)$ is a weak strip then $\nu/\eta$ is a horizontal strip, we can write $h_{\lambda}=s^{(k)}_{\lambda}+\sum_{\lambda\lhd\mu\in\mathcal{P}_{k}}K^{(k)}_{\mu\lambda}s^{(k)}_{\mu}$ for $\lambda\in\mathcal{P}_{k}$ by the same argument as the case of Schur functions, which ensures the well-definedness of $s^{(k)}_{\lambda}$ and shows that $\{s^{(k)}_{\lambda}\}_{\lambda\in\mathcal{P}_{k}}$ forms a basis of $\Lambda^{(k)}=\mathbb{Z}[h_{1},\dots,h_{k}]\subset\Lambda$ . In addition $s^{(k)}_{\lambda}$ is homogeneous of degree $|\lambda|$ .

Note that $s^{(k)}_{(r)}=h_{r}$ for $1\leq r\leq k$ since $\mathfrak{c}(\lambda)/\varnothing$ is a weak $r$ -strip if and only if $\lambda=(r)$ . In [MR2331242, Property 39] it is proved that if $\lambda_{1}+l(\lambda)\leq k+1$ (in other words $\lambda\subset R_{t}$ for some $t$ ) then $s^{(k)}_{\lambda}=s_{\lambda}$ .

It is proved in [MR2331242, Theorem 40] that

Proposition 5 ( $k$ -rectangle property).

For $1\leq t\leq k$ and $\lambda\in\mathcal{P}_{k}$ , we have $s^{(k)}_{R_{t}\cup\lambda}=s^{(k)}_{R_{t}}s^{(k)}_{\lambda}(=s_{R_{t}}s^{(k)}_{\lambda}).$

2.6. $K$ - $k$ -Schur functions $g^{(k)}_{\lambda}$

In [Morse12] a combinatorial characterization of $K$ - $k$ -Schur functions is given via an analogue of the Pieri rule, using some kind of strips called affine set-valued strips.

For a partition $\lambda$ , $(i,j)\in(\mathbb{Z}_{>0})^{2}$ is called $\lambda$ -blocked if $(i+1,j)\in\lambda$ .

Definition 6 (affine set-valued strip).

For $r\leq k$ , $(\gamma/\beta,\rho)$ is called an affine set-valued strip of size $r$ (or an affine set-valued $r$ -strip) if $\rho$ is a partition and $\beta\subset\gamma$ are cores both containing $\rho$ such that

(1)

$\gamma/\beta$ * is a weak $(r-m)$ -strip where we put $m=\#\mathrm{Res}(\beta/\rho)$ ,* 2. (2)

$\beta/\rho$ * is a subset of $\beta$ -removable corners,* 3. (3)

$\gamma/\rho$ * is a horizontal strip,* 4. (4)

For all $i\in\mathrm{Res}(\beta/\rho)$ , all $\beta$ -removable $i$ -corners which are not $\gamma$ -blocked are in $\beta/\rho$ .

In this paper we employ the following characterization [Morse12, Theorem 48] of the $K$ - $k$ -Schur function as its definition.

Definition 7 ( $K$ - $k$ -Schur function via an “affine set-valued” Pieri rule).

$K$ - $k$ -Schur functions $\{g^{(k)}_{\lambda}\}_{\lambda\in\mathcal{P}_{k}}$ are the family of symmetric functions such that $g^{(k)}_{\varnothing}=1$ and for $\lambda\in\mathcal{P}_{k}$ and $0\leq r\leq k$ ,

[TABLE]

summed over $(\mu,\rho)$ such that $(\mathfrak{c}(\mu)/\mathfrak{c}(\lambda),\rho)$ is an affine set-valued strip of size $r$ .

Notice that, given a weak strip $\gamma/\beta$ , taking a $\rho$ such that $(\gamma/\beta,\rho)$ becomes an affine set-valued strip is equivalent to choosing a subset of the set of residues $i\in\mathbb{Z}/(k+1)$ where there is at least one $\gamma$ -nonblocked $\beta$ -removable $i$ -corner.

Now we introduce a notation for the convinience:

Definition 8.

For partitions $\lambda,\mu$ , we denote by $r_{\lambda\mu}$ the number of distinct residues of $\lambda$ -nonblocked $\mu$ -removable corners.

Then for a fixed weak $(r-m)$ -strip $\gamma/\beta$ , the number of $\rho$ such that $(\gamma/\beta,\rho)$ is an affine set-valued $r$ -strip is equal to $\displaystyle\binom{r_{\gamma\beta}}{m}$ . Notice that $\gamma/\beta$ with all $\gamma$ -nonblocked $\beta$ -removable corners added is a horizontal strip. Therefore we can rewrite Definition 7:

Proposition 9.

For $\lambda\in\mathcal{P}_{k}$ and $0\leq r\leq k$ ,

[TABLE]

We can prove similarly that $g^{(k)}_{\lambda}$ is uniquely determined by (7): for $\lambda\in\mathcal{P}_{k}$ and $1\leq r\leq k$ , we have $h_{r}g^{(k)}_{\lambda}=g^{(k)}_{\lambda\cup(r)}+\sum_{\mu}a_{\mu}g^{(k)}_{\mu}$ with $\mu\in\mathcal{P}_{k}$ satisfying $|\mu|<|\lambda\cup(r)|$ or $\mu\rhd\lambda\cup(r)$ . Thus, for $\lambda\in\mathcal{P}_{k}$ we can write $h_{\lambda}=g^{(k)}_{\lambda}+\sum_{\mu}\mathcal{K}^{(k)}_{\mu\lambda}g^{(k)}_{\mu}$ , summed over $\mu\in\mathcal{P}_{k}$ satifying $|\mu|<|\lambda|$ or $\mu\rhd\lambda$ . Hence $g^{(k)}_{\lambda}$ is well-defined and $\{g^{(k)}_{\lambda}\}_{\lambda\in\mathcal{P}_{k}}$ forms a basis of $\Lambda^{(k)}$ .

Note that $g^{(k)}_{(r)}=h_{r}$ for $1\leq r\leq k$ since if $(\mathfrak{c}(\mu)/\varnothing,\rho)$ is an affine set-valued $r$ -strip then $(\mu,\rho)=((r),\varnothing)$ . Moreover, though $g^{(k)}_{\lambda}$ is an inhomogeneous symmetric function in general, from the form of (7) we can deduce that the degree of $g^{(k)}_{\lambda}$ is $|\lambda|$ and its homogeneous part of highest degree is equal to $s^{(k)}_{\lambda}$ by using induction.

2.7. Some properties of bounded partitions and cores

In this section we review some properties which show that the $k$ -rectangles $R_{t}=(t^{k+1-t})$ are important and thus it can be expected that there are some good properties of $g^{(k)}_{\lambda}$ ’s where $\lambda$ can be written in the form $\lambda=R_{t}\cup\mu$ .

Recall the weak order $\prec$ of Definition 2.

Corollary 10.

For $\mu,\lambda\in\mathcal{P}_{k}$ and $P=R_{t_{1}}^{a_{1}}\cup\cdots\cup R_{t_{m}}^{a_{m}}$ $($$1\leq t_{1}<\cdots<t_{m}\leq k$ and $a_{1},\dots,a_{m}\in\mathbb{Z}_{>0}$$)$ ,

[TABLE]

Proof.

See [MR2079931, Theorem 20] for the case where $m=1$ and $a_{1}=1$ (i.e. $P=R_{t_{1}}$ ). The general case follows by using this case repeatedly. ∎

Proposition 11.

For $\nu,\lambda\in\mathcal{P}_{k}$ and $P=R_{t_{1}}^{a_{1}}\cup\cdots\cup R_{t_{m}}^{a_{m}}$ $($$1\leq t_{1}<\cdots<t_{m}\leq k$ and $a_{1},\dots,a_{m}\in\mathbb{Z}_{>0}$$)$ ,

[TABLE]

Proof.

The case where $m=1$ and $a_{1}=1$ (i.e. $P=R_{t_{1}}$ ) is proved in the proof of [MR2331242, Theorem 40]. The general case follows by using this case repeatedly. ∎

Corollary 12.

For $\eta,\lambda\in\mathcal{P}_{k}$ and $P=R_{t_{1}}^{a_{1}}\cup\cdots\cup R_{t_{m}}^{a_{m}}$ $($$1\leq t_{1}<\cdots<t_{m}\leq k$ and $a_{1},\dots,a_{m}\in\mathbb{Z}_{>0}$$)$ ,

[TABLE]

Proof.

$\Longrightarrow$ : We have $\lambda\cup P\preceq\mu$ by the definition of weak strips. Thus we can write $\mu=\exists\nu\cup P$ by Proposition 10. Then we have that $\mathfrak{c}(\nu)/\mathfrak{c}(\lambda)$ is a weak strip by Proposition 11.

$\Longrightarrow$ : By Proposition 11. ∎

Lemma 13.

Let $\lambda\in\mathcal{P}_{k}$ , $1\leq t\leq k$ , and let $r\in\mathbb{Z}_{\geq 0}$ such that $\lambda_{r}\geq t\geq\lambda_{r+1}$ , where we regard $\lambda_{0}=\infty$ . Put $\tilde{\lambda}=\lambda\cup R_{t}$ . Then

[TABLE]

Proof.

The latter case is obvious since $\tilde{\lambda}_{i+(k+1-t)}=\lambda_{i}$ for $i\geq r+1$ .

For $i=r+(k+1-t),\ldots,r+1$ ,

[TABLE]

Then for $i=r,r-1,\ldots,1$ , by descending induction on $i$ ,

[TABLE]

∎

Remark. There are more than one candidates for $r$ if $\lambda$ has a part equal to $t$ , thus in such situations both equalities of the above lemma may hold for some $i$ .

3. Possibility of factoring out $g^{(k)}_{R_{t_{1}}^{a_{1}}\cup\dots\cup R_{t_{m}}^{a_{m}}}$ and some other general results

Recall how to prove the formula $s^{(k)}_{R_{t}\cup\lambda}=s^{(k)}_{R_{t}}s^{(k)}_{\lambda}$ in [MR2331242]: first consider a linear map $\Theta$ extending $s^{(k)}_{\lambda}\mapsto s^{(k)}_{R_{t}\cup\lambda}$ for all $\lambda\in\mathcal{P}_{k}$ . Then from the weak Pieri rule it was shown that it commutes with the multiplication by $h_{r}$ , and thus that $\Theta$ coincides with the multiplication by $s^{(k)}_{R_{t}}$ . In the case of $K$ - $k$ -Schur functions, a similar map $\Theta$ does not commute with the multiplication of $h_{r}$ since the Pieri rule is different in lower terms. However, it holds that $g^{(k)}_{R_{t}}$ divides $g^{(k)}_{R_{t}\cup\lambda}$ . We prove it in a slightly more general form.

The following notation is often referred later:

$(\mathsf{NP})$

Let $1\leq t_{1},\dots,t_{m}\leq k$ be distinct integers and $a_{i}\in\mathbb{Z}_{>0}\ (1\leq i\leq m)$ , where $m\in\mathbb{Z}_{>0}$ . Then we put

[TABLE]

Proposition 14.

Let $P$ be as in the above $(\mathsf{NP})$ . Then, for $\lambda=(\lambda_{1},\cdots,\lambda_{l})\in\mathcal{P}_{k}$ , we have $g^{(k)}_{P}|g^{(k)}_{\lambda\cup P}$ in the ring $\Lambda^{(k)}$ .

Remark. Note that $\lambda$ may still have the form $\lambda=R_{t}\cup\mu$ . Hereafter we will not repeat the same remark in similar statements.

Proof.

we prove it by induction on $\lambda$ , with respect to the order $\leq$ defined by $\mu\leq\lambda\iff|\mu|<|\lambda|$ or ( $|\mu|=|\lambda|$ and $\mu\rhd\lambda$ ). The statement is obvious when $\lambda=\varnothing$ .

Assume $\lambda\neq\varnothing$ and put $\hat{\lambda}=(\lambda_{1},\cdots,\lambda_{l-1})$ . Then

[TABLE]

for some coefficients $a_{\lambda\mu}$ , since adding a weak strip to $P\cup\hat{\lambda}$ yields a $k$ -bounded partition in the form of $P\cup\mu$ for some $\mu\in\mathcal{P}_{k}$ , by Proposition 10. Here $\mu$ in the summation runs under the condition $|\mu|<|\lambda|$ or $\mu\rhd\lambda$ . By induction hypothesis $g^{(k)}_{P\cup\hat{\lambda}}$ and $g^{(k)}_{P\cup\mu}$ are divisible by $g^{(k)}_{P}$ if $|\mu|<|\lambda|$ or $\mu\rhd\lambda$ . This completes the proof. ∎

Since the homogeneous part of highest degree of $g^{(k)}_{\lambda}$ is equal to $s^{(k)}_{\lambda}$ for any $\lambda$ , it follows from Propositions 5 and 14 that

Corollary 15.

Let $P$ be as in $(\mathsf{NP})$ . Then, for any $\lambda\in\mathcal{P}_{k}$ , we can write

[TABLE]

summing over $\mu\in\mathcal{P}_{k}$ such that $|\mu|<|\lambda|$ , for some coefficients $a_{\lambda\mu}$ (depending on $P$ ).

Now we are interested in finding a explicit description of $g^{(k)}_{P\cup\lambda}\big{/}g^{(k)}_{P}$ . Let us consider the case $P=R_{t}$ for simplicity.

As noted above, a linear map $\Theta$ extending $g^{(k)}_{\lambda}\mapsto g^{(k)}_{R_{t}\cup\lambda}$ ( $\forall\lambda\in\mathcal{P}_{k}$ ) does not coincide with the multiplication of $g^{(k)}_{R_{t}}$ because it does not commute with the multiplication by $h_{r}$ in the first place.

However, in the remaining part of this section, we can prove that the restriction of $\Theta$ to the subspace spanned by $\{g^{(k)}_{R_{t}\cup\mu}\}_{\mu\in\mathcal{P}_{k}}$ (in fact this is the principal ideal generated by $g^{(k)}_{R_{t}}$ ) commutes with the multiplication by $h_{r}$ , and thus it coincides with the multiplication of $\Theta(g^{(k)}_{R_{t}})\big{/}g^{(k)}_{R_{t}}=g^{(k)}_{R_{t}\cup R_{t}}\big{/}g^{(k)}_{R_{t}}$ on that ideal (Proposition 18). Thus it is of interest to describe the value of $g^{(k)}_{R_{t}\cup R_{t}}\big{/}g^{(k)}_{R_{t}}$ , which is shown to be $\sum_{\nu\subset R_{t}}g^{(k)}_{\nu}$ in the author’s following paper [Takigiku].

Now let us begin with seeing how $\Theta$ and the multiplication by $h_{r}$ do not commute. Recall the $K$ - $k$ -Schur version of the Pieri rule (7)

[TABLE]

and compare with the formula obtained by replacing $\lambda$ with $R_{t}\cup\lambda$ :

[TABLE]

By Corollary 12, the summation in (8) is formed for all $\eta$ having the form $\eta=R_{t}\cup\nu$ such that $\mathfrak{c}(\nu)/\mathfrak{c}(\lambda)$ is a weak $s$ -strip. Hence the right-hand side of (8) differs from what is obtained by replacing each $\nu$ in the right-hand side of (7) by $R_{t}\cup\nu$ according to the difference between $r_{\mathfrak{c}(\nu)\mathfrak{c}(\lambda)}$ and $r_{\mathfrak{c}(R_{t}\cup\nu)\mathfrak{c}(R_{t}\cup\lambda)}$ .

The next lemma says $r_{\mathfrak{c}(R_{t}\cup\nu)\mathfrak{c}(R_{t}\cup\lambda)}=r_{\mathfrak{c}(\nu)\mathfrak{c}(\lambda)}$ holds if $\lambda$ has a part equal to $t$ .

Lemma 16.

For $\nu,\lambda\in\mathcal{P}_{k}$ such that $\lambda$ has a part equal to $t$ and $\mathfrak{c}(\nu)/\mathfrak{c}(\lambda)$ is a weak strip, we have $r_{\mathfrak{c}(\nu),\mathfrak{c}(\lambda)}=r_{\mathfrak{c}(\nu\cup R_{t}),\mathfrak{c}(\lambda\cup R_{t})}$ .

Proof.

We write $\tilde{\lambda}=\lambda\cup R_{t}$ and $\tilde{\nu}=\nu\cup R_{t}$ . We take $r$ such that $\lambda_{r}=t>\lambda_{r+1}$ (then $\nu_{r}\geq t=\lambda_{r}\geq\nu_{r+1}$ since $\nu/\lambda$ is a horizontal strip). Then we have

[TABLE]

therefore by Lemma 13

[TABLE]

(here we applied Lemma 13 to $\lambda$ and $r-1$ for the lower equation) and

[TABLE]

Then,

(1)

if $i<r+(k+1-t)$ ,

$(i,\mathfrak{c}(\tilde{\lambda})_{i})$ is a $\mathfrak{c}(\tilde{\lambda})$ -removable corner $\iff$ $(i,\mathfrak{c}(\lambda)_{i})$ is a $\mathfrak{c}(\lambda)$ -removable corner, 2. (2)

if $i\geq r+(k+1-t)$ ,

$(i,\mathfrak{c}(\tilde{\lambda})_{i})$ is a $\mathfrak{c}(\tilde{\lambda})$ -removable corner $\iff$ $(i-(k+1-t),\mathfrak{c}(\lambda)_{i-(k+1-t)})$ is a $\mathfrak{c}(\lambda)$ -removable corner.

Moreover, when $(i,\mathfrak{c}(\tilde{\lambda})_{i})$ is a $\mathfrak{c}(\tilde{\lambda})$ -removable corner (of residue $a$ ), we consider two cases:

(1)

if $i<r+(k+1-t)$ . Then

[TABLE]

and the residue of $(i,\mathfrak{c}(\lambda)_{i})$ is $a-t$ . 2. (2)

if $i\geq r+(k+1-t)$ . Then

[TABLE]

and the residue of $(i-(k+1-t),\mathfrak{c}(\lambda)_{i-(k+1-t)})$ is $a-t$ .

Hence, for each $a\in\mathbb{Z}/(k+1)$ , there exists a non- $\mathfrak{c}(\tilde{\nu})$ -blocked $\mathfrak{c}(\tilde{\lambda})$ -removable $a$ -corner if and only if there exists a non- $\mathfrak{c}(\nu)$ -blocked $\mathfrak{c}(\lambda)$ -removable $(a-t)$ -corner. Therefore we have $r_{\mathfrak{c}(\nu)\mathfrak{c}(\lambda)}=r_{\mathfrak{c}(\tilde{\nu})\mathfrak{c}(\tilde{\lambda})}$ . ∎

As a corollary of the proof of the above lemma, we have

Corollary 17.

For any $\lambda,\nu\in\mathcal{P}_{k}$ and $1\leq t\leq k$ we have $r_{\mathfrak{c}(R_{t}\cup\nu)\mathfrak{c}(R_{t}\cup\lambda)}=r_{\mathfrak{c}(\nu)\mathfrak{c}(\lambda)}$ or $r_{\mathfrak{c}(\nu)\mathfrak{c}(\lambda)}+1$ .

Proof.

Take $r$ such that $\lambda_{r}\geq t>\lambda_{r+1}$ and do a same argument as the above lemma.

Then we have that, if $i\neq r+(k+1-t)$ , there exists a $\mathfrak{c}(\tilde{\nu})$ -nonblocked $\mathfrak{c}(\tilde{\lambda})$ -removable $a$ -corner in $i$ -th row if and only if there exists a $\mathfrak{c}(\nu)$ -nonblocked $\mathfrak{c}(\lambda)$ -removable $(a-t)$ -corner in $i^{\prime}$ -th row. (Here we put $i^{\prime}=i$ if $i<r+(k+1-t)$ and $i^{\prime}=i-(k+1-t)$ if $i>r+(k+1-t)$ )

Hence we have $r_{\mathfrak{c}(\nu)\mathfrak{c}(\lambda)}\leq r_{\mathfrak{c}(\tilde{\nu})\mathfrak{c}(\tilde{\lambda})}\leq r_{\mathfrak{c}(\nu)\mathfrak{c}(\lambda)}+1$ . ∎

Proposition 18.

For $\lambda\in\mathcal{P}_{k}$ and $1\leq t\leq k$ , we have $g^{(k)}_{\lambda\cup R_{t}\cup R_{t}}=g^{(k)}_{\lambda\cup R_{t}}\cdot\displaystyle\frac{g^{(k)}_{R_{t}\cup R_{t}}}{g^{(k)}_{R_{t}}}$ .

Proof.

Write $\tilde{\mu}=\mu\cup R_{t}$ for $\mu\in\mathcal{P}_{k}$ .

Define a linear map $\Theta:\Lambda^{(k)}\longrightarrow\Lambda^{(k)}$ by $g^{(k)}_{\mu}\longmapsto g^{(k)}_{\tilde{\mu}}$ for all $\mu\in\mathcal{P}_{k}$ and put $X=\mathrm{span}\{g^{(k)}_{\tilde{\lambda}}\mid\lambda\in\mathcal{P}_{k}\}$ . Then $X$ is an ideal of $\Lambda^{(k)}$ because $h_{r}\cdot g^{(k)}_{\tilde{\lambda}}$ can be written as a linear combination of $\{g^{(k)}_{\tilde{\nu}}\mid\nu\in\mathcal{P}_{k}\}$ , by (7) and Proposition 12.

Next we claim

[TABLE]

for $1\leq r\leq k$ , where $h_{r}\cdot$ denotes the multiplication by $h_{r}$ .

Proof of claim.

It suffices to show $h_{r}\cdot g^{(k)}_{\widetilde{\mu\cup R_{t}}}=\Theta(h_{r}\cdot g^{(k)}_{\tilde{\mu}})$ for $\mu\in\mathcal{P}_{k}$ . More generally, we can show $h_{r}\cdot g^{(k)}_{\widetilde{\mu\cup(t)}}=\Theta(h_{r}\cdot g^{(k)}_{\mu\cup(t)})$ for $\mu\in\mathcal{P}_{k}$ :

[TABLE]

Here the second equality uses Proposition 12, and the third equality uses Lemma 16. Hence the claim is proved.

Since $h_{1},\ldots,h_{k}$ generate $\Lambda^{(k)}$ , the claim implies that $\Theta|_{X}$ is a $\Lambda^{(k)}$ -module homomorphism. Hence for any $x\in X$ ,

[TABLE]

which implies $\Theta(x)=x\cdot\displaystyle\frac{g^{(k)}_{R_{t}\cup R_{t}}}{g^{(k)}_{R_{t}}}$ for any $x\in X$ . Setting $x=g^{(k)}_{R_{t}\cup\lambda}$ gives the proposition. ∎

Theorem 19.

Let $P=R_{t_{1}}^{a_{1}}\cup\dots\cup R_{t_{m}}^{a_{m}}$ be as in $(\mathsf{NP})$ , and put $Q=R_{t_{1}}\cup\cdots\cup R_{t_{m}}$ . Then, for $\lambda\in\mathcal{P}_{k}$ we have

[TABLE]

Proof.

Induction on $\sum_{i}(a_{i}-1)$ . If $\sum_{i}(a_{i}-1)=0$ then it is obvious since $P=Q$ .

Otherwise, we can assume $a_{1}>1$ without loss of generality. Write $P=R_{t_{1}}\cup R_{t_{1}}\cup P^{\prime}$ . By Proposition 18 we have

[TABLE]

thus we conclude

[TABLE]

Here we used induction hypothesis for the second equality. ∎

4. A factorization of $g^{(k)}_{R_{t_{1}}^{a_{1}}\cup\dots\cup R_{t_{m}}^{a_{m}}\cup(r)}$

In this section we will give an explicit formula for $g^{(k)}_{R_{t_{1}}^{a_{1}}\cup\dots\cup R_{t_{m}}^{a_{m}}\cup\lambda}\big{/}g^{(k)}_{R_{t_{1}}^{a_{1}}\cup\dots\cup R_{t_{m}}^{a_{m}}}$ when $\lambda=(r)$ .

Roughly speaking, $K$ - $k$ -Schur functions can be calculated by “solving” the system of Pieri rule formulas (7). To solve such a system, it is important to understand concretely what weak strips $\mathfrak{c}(\nu)/\mathfrak{c}(\mu)$ are.

If $\mu$ is a union of $k$ -rectangles $P=R_{t_{1}}^{a_{1}}\cup\dots\cup R_{t_{m}}^{a_{m}}$ the situation is simple: if $\mathfrak{c}(\nu)/\mathfrak{c}(P)$ is a weak strip then $\nu$ has the form $P\cup(s)$ for some $s$ , as we will see in the proof of the following proposition. Thus the Pieri rule also has a simple explicit expression as follows:

Proposition 20.

Let $P$ and $\alpha_{P}(u)$ $(u\in\mathbb{Z}_{>0})$ be as in $(\mathsf{NP})$ in Section 3, before Proposition 14. Then, for $1\leq r\leq k$ , we have

[TABLE]

Proof.

We have $\mathfrak{c}(P)=\underbrace{R_{t_{m}}\oplus\dots\oplus R_{t_{m}}}_{a_{m}}\oplus\cdots\oplus\underbrace{R_{t_{1}}\oplus\dots\oplus R_{t_{1}}}_{a_{1}}$ and all addable corners of $\mathfrak{c}(P)$ has the same residue, say $i$ . Moreover, $\mathfrak{c}(P)$ has a total of $\sum_{j}a_{j}$ removable corners, $a_{j}$ of which are derived from the removable corner of $R_{t_{j}}$ and having the residue $i+t_{j}$ for each $j$ .

Next we claim that if $\gamma/\mathfrak{c}(P)$ is a weak $s$ -strip then $\gamma=s_{i+s-1}\cdots s_{i+1}s_{i}(\mathfrak{c}(P))$ .

Proof of the claim. We prove it by induction on $s$ . If $s=1$ , it is obvious because all addable corners of $\mathfrak{c}(P)$ have the same residue $i$ .

Let $s>1$ and $\gamma/\mathfrak{c}(P)$ be a weak $s$ -strip. Then we can write $\gamma=s_{j_{s}}\cdots s_{j_{2}}s_{j_{1}}(\mathfrak{c}(P))$ , where $(j_{s},\cdots,j_{1})$ is cyclically decreasing (see Definition-Proposition 3(4)).

Since $s_{j_{s-1}}\cdots s_{j_{2}}s_{j_{1}}(\mathfrak{c}(P))/\mathfrak{c}(P)$ is a weak $(s-1)$ -strip, we have $(j_{s-1},\cdots,j_{1})=(i+s-2,\cdots,i+1,i)$ by the induction hypothesis. Since $(j_{s},i+s-2,\cdots,i+1,i)$ is cyclycally decreasing, we have $j_{s}\notin\{i-1,i,i+1,\cdots,i+s-2\}$ .

If $j_{s}\neq i+s-1$ , then $s_{j_{s}}$ commutes with $s_{i},s_{i+1},\cdots,s_{i+s-2}$ and

[TABLE]

However, $|s_{j_{s}}(\mathfrak{c}(P))|_{k}\leq|\mathfrak{c}(P)|_{k}$ because $\mathfrak{c}(P)$ doesn’t have an addable corner of residue $j_{s}$ . Hence $|\gamma|_{k}\leq|\mathfrak{c}(P)|_{k}+s-1$ , violating the assumption that $\gamma/\mathfrak{c}(P)$ is a weak $s$ -strip.

Hence we have $j_{s}=i+s-1$ , completing the proof of the claim.

Since $s_{i+s-1}\cdots s_{i+1}s_{i}(\mathfrak{c}(P))$ has the form below on the right, we can see that the corresponding $k$ -bounded partition has the form $P\cup(s)$ .

Now we get back to the proof of the proposition. Let $\gamma=s_{i+s-1}\cdots s_{i+1}s_{i}(\mathfrak{c}(P))$ . Then the removable corner of $\mathfrak{c}(P)$ corresponding to the removable corner of $R_{t_{a}}$ is $\gamma$ -blocked if and only if $s\geq t_{a}$ . Then the number of residues of $\gamma$ -nonblocked removable corners of $\mathfrak{c}(P)$ is exactly $\alpha_{P}(s+1)$ . ∎

The above proposition gives an expression for $g^{(k)}_{P}h_{r}$ as a linear combination of $\{g^{(k)}_{P\cup(s)}\}_{s}$ . To solve this linear equation, we need the following lemma of binomial coefficients.

Lemma 21.

Let $l$ be a positive integer and $\beta_{1},\beta_{2},\cdots,\beta_{l+1}$ be integers such that $\beta_{i}\geq\beta_{i+1}\geq\beta_{i}-1$ for each $i$ .

Let $C=\bigg{(}(-1)^{r-s}\displaystyle\binom{\beta_{s+1}}{r-s}\bigg{)}_{r,s=0}^{l}$ . Then $C^{-1}=\displaystyle\bigg{(}\binom{\beta_{r}+r-s-1}{r-s}\bigg{)}_{r,s=0}^{l}$ .

Here $\displaystyle\binom{a}{b}$ is considered to be [math] if $b<0$ .

Proof.

Put $D=\displaystyle\bigg{(}\binom{\beta_{r}+r-s-1}{r-s}\bigg{)}_{r,s=0}^{l}$ . The $(p,q)$ element of the matrix $DC$ is

[TABLE]

which is 0 unless $p\geq q$ .

Let us consider the case $p\geq q$ .

By applying the next lemma for $a=\beta_{q+1},\ b=\beta_{p},\ c=p-q\geq 0$ , we have

[TABLE]

where the last equality follows from $\beta_{q+1}-\beta_{p}\in\{0,1,\cdots,p-q-1\}$ (if $q+1\leq p$ ). ∎

Lemma 22.

For integers $a$ , $b$ and a nonnegative integer $c$ ,

[TABLE]

Proof.

Since $\binom{m}{n}$ is the coefficient of $X^{n}$ in $(1+X)^{m}\in\mathbb{Z}[[X]]$ for $m\in\mathbb{Z}$ and $n\in\mathbb{Z}_{\geq 0}$ , we have

[TABLE]

∎

Now we can deduce the formula showing the goal of this section.

Theorem 23.

If $P,\alpha_{P}(u)$ and $r$ are as in Proposition 20, then we have

[TABLE]

In particular, if $t_{m}<r$ , which means $\alpha_{P}(r)=0$ , we have

[TABLE]

On the other hand, when $m=1$ ,

[TABLE]

Proof.

Apply Lemma 21 for Proposition 20. ∎

5. A factorization of $g^{(k)}_{R_{t_{1}}^{a_{1}}\cup\dots\cup R_{t_{m}}^{a_{m}}\cup\lambda}$ with small $\lambda$ and splitting $g^{(k)}_{R_{t_{1}}^{a_{1}}\cup\dots\cup R_{t_{m}}^{a_{m}}}$ into $g^{(k)}_{R_{t_{1}}^{a_{1}}}\dots g^{(k)}_{R_{t_{m}}^{a_{m}}}$

5.1. Statements

Our goal in this section is to show the equality

[TABLE]

for $1\leq t_{1}<\cdots<t_{m}\leq k$ and $a_{i}>0$ (see Theorem 31).

The essential part is to prove $g^{(k)}_{R_{t_{1}}\cup\dots\cup R_{t_{m}}}=g^{(k)}_{R_{t_{1}}\cup\dots\cup R_{t_{m-1}}}g^{(k)}_{R_{t_{m}}}$ , and the remainging part follows from the results from Section 3 and induction (on $n$ ). This is a simple statement, but our proof involves an induction on the shape of partitions, thus we have to prove a more general statement (see the case $t_{n}<r$ of part (2) of Theorem 30): Let $P=\bigcup_{i=1}^{m}R_{t_{i}}^{a_{i}}$ be as in $(\mathsf{NP})$ , Section 3, before Proposition 14, and $\lambda$ as follows:

$\mathsf{(N\lambda)}$

Let $(\varnothing\neq)\lambda\in\mathcal{P}_{k}$ with satisfying ${\bar{\lambda}}\subset R^{\prime}_{{\bar{l}}}$ , where we write ${\bar{\lambda}}=(\lambda_{1},\lambda_{2},\dots,\lambda_{l(\lambda)-1})$ and ${\bar{l}}=l({\bar{\lambda}})=l(\lambda)-1$ . (Here we consider $R_{t}$ to be $\varnothing$ unless $1\leq t\leq k$ )

(Note: when $l(\lambda)=1$ , we have ${\bar{l}}=0$ and ${\bar{\lambda}}=\varnothing=R^{\prime}_{{\bar{l}}}$ thus $\lambda$ satisfies $\mathsf{(N\lambda)}$ . When $l(\lambda)>k+1$ , we have ${\bar{l}}>k$ and ${\bar{\lambda}}\neq\varnothing=R^{\prime}_{{\bar{l}}}$ thus $\lambda$ does not satisfy $\mathsf{(N\lambda)}$ . )

Then,

[TABLE]

5.2. Proofs

We will prove a slightly even more general formula than (9) (see part (2) of Theorem 30) in the following procedure.

•

Step (A)**:

First we write $g^{(k)}_{\lambda}$ as a linear combination of products of $h_{i}$ ’s and $g^{(k)}_{\mu}$ ’s with $l(\mu)<l(\lambda)$ : putting $\bar{\lambda}=(\lambda_{1},\dots,\lambda_{l(\lambda)-1})$ , we have

[TABLE]

if $\bar{\lambda}_{1}+l(\bar{\lambda})\leq k+1$ (Lemmas 26 (1), 28 (1), 29 (1) and 34).

•

Step (B)**:

Derive a similar expression for $g^{(k)}_{P\cup\lambda}$ (parts Lemmas 26 (2)-(3), 28 (2)-(3), 29 (2)-(3), 34).

•

Step (C)**:

Compare (B) with the equality obtained by multiplying the formula in Step (A) by $g^{(k)}_{P}$ , noticing $g^{(k)}_{P}g^{(k)}_{\mu}=g^{(k)}_{P\cup\mu}$ by induction.

Step (A) consists of two substeps:

•

Step (A-1): Write down the Pieri rule for $g^{(k)}_{\mu}h_{r}$ explicitly.

•

Step (A-2): Solve the system of Pieri rule formulas to give expressions for $g^{(k)}_{\lambda}$ as a linear combination of $\{g^{(k)}_{\mu}h_{r}\}_{\mu,r}$ .

Obtaining an expression for $g^{(k)}_{P\cup\lambda}$ in Step (B) follows from similar steps (B-1) and (B-2).

•

Step (B-1): Write down the Pieri rule for $g^{(k)}_{P\cup\mu}h_{r}$ explicitly.

•

Step (B-2): Solve the system of Pieri rule formulas to give expressions for $g^{(k)}_{P\cup\lambda}$ as a linear combination of $\{g^{(k)}_{P\cup\mu}h_{r}\}_{\mu,r}$ .

5.3. Steps (A-1) and (B-1)

Toward Step (A-1) and (B-1), let us begin with describing weak strips $\mathfrak{c}(\lambda)/\mathfrak{c}(\mu)$ where $\mu$ is contained in a $k$ -rectangle.

Lemma 24.

Assume $\mu\subset R_{k+1-l}$ and $\mu_{l}>0$ . Let $0\leq u\leq\mu_{l}$ be an integer.

$(1)$ * For $\kappa\in\mathcal{P}_{k}$ ,*

[TABLE]

$(2)$ * For $\tilde{\kappa}\in\mathcal{P}_{k}$ ,*

[TABLE]

Proof.

(1): The second equivalence is obvious.

The “if” part of the first equivalence is easy: since $\kappa_{1}\leq k+1-l$ and $l(\kappa)\leq l+1$ , we have $\mathfrak{c}(\kappa)=\kappa$ or $\mathfrak{c}(\kappa)=(\kappa_{1}+\kappa_{l+1},\kappa_{2},\ldots)$ and hence ${\kappa}^{\omega_{k}}/{\mu}^{\omega_{k}}$ is a vertical strip.

Hence it suffices to prove the “only if” part of the first equivalence: let $\mu\subset R_{k+1-l}$ , $\mu_{l}>0$ , and $\mathfrak{c}(\kappa)/\mathfrak{c}(\mu)$ be a weak strip of size $\leq\mu_{l}$ , and we shall prove $\kappa_{1}\leq k+1-l$ .

[TABLE]

Assume, on the contrary, that $\kappa_{1}>k+1-l$ . Write $\bar{\kappa}=(\kappa_{2},\kappa_{3},\dots)\subset R_{k+1-l}$ . Then by Lemma 1 we have

[TABLE]

Hence, the hook lengths of $(1,1),\cdots,(1,\mu_{l})$ in $\mathfrak{c}(\kappa)$ are all greater than $k$ because

[TABLE]

for $1\leq j\leq\mu_{l}$ . On the other hand, those of $(2,1),\cdots,(2,\mu_{l})$ in $\mathfrak{c}(\kappa)$ are less than or equal to $k$ because $\bar{\kappa}\subset\mu\subset R_{k+1-l}$ .

Hence

[TABLE]

for $1\leq j\leq\mu_{l}$ .

Since $\mathfrak{c}(\kappa)/\mathfrak{c}(\mu)$ is a weak strip, ${\kappa}^{\omega_{k}}/{\mu}^{\omega_{k}}$ is a vertical strip. Hence

[TABLE]

for $1\leq j\leq\mu_{l}$ , which implies $\kappa_{l+1}\geq\mu_{l}$ . Then we have

[TABLE]

since $\kappa_{1}>k+1-l\geq\mu_{1}$ . This is a contradiction.

(2): The first equivalence follows from Corollary 12. The second equivalence follows from (1). ∎

Next let us explicitly describe the weak Pieri rule (7), after we prepare a notation for convenience.

Definition 25.

Let $P$ and $\alpha_{P}(u)$ $(u\in\mathbb{Z}_{>0})$ be as in $(\mathsf{NP})$ . For $\nu\subset R_{k+1-l(\nu)}$ , $0\leq u\leq\nu_{l(\nu)}$ , $p\in\mathbb{Z}$ , we set

[TABLE]

Lemma 26.

Let $P$ and $\alpha_{P}(u)$ $(u\in\mathbb{Z}_{>0})$ be as in $(\mathsf{NP})$ . Assume $\mu\subset R_{k+1-l}$ , $\mu_{l}>0$ , $\mu_{l}\geq r\geq 0$ . Then we have

$(1)$ **

[TABLE]

$(2)$ * If $\max_{i}(t_{i})<\mu_{l}$ ,*

[TABLE]

$(3)$ * If $\max_{i}(t_{i})=\mu_{l}$ ,*

[TABLE]

Proof.

(1) We transform the right-hand side of Eq. (7), Proposition 9, into the right-hand side of part (1) of the Lemma as follows:

[TABLE]

Here, the equality (i) uses Lemma 24 (1) in order to change the summation variable from $\kappa$ to $\nu$ according to $\kappa=\nu\cup(u-|\nu/\mu|)$ .

For the equality (ii) we use (1) of the following Lemma 27 and put $s=r-u$ . Note that $u-|\nu/\mu|\geq\mu_{l}$ occurs only if $u-|\nu/\mu|=u=r=\mu_{l}$ since $u\leq r\leq\mu_{l}$ , in which case we have $\displaystyle\binom{r_{\nu\cup(u-|\nu/\mu|),\mu}}{r-u}=1=\displaystyle\binom{r_{\nu,\mu}}{r-u}$ .

We can prove (2) and (3) almost the same as (1), using Lemma 24 (2) for (i), and (2) of the following Lemma 27 for (ii).

Note that, in the same way as (1), the case $u-|\nu/\mu|\geq\mu_{l}$ and $\max_{i}(t_{i})<\mu_{l}$ appears in the expression for $g^{(k)}_{P\cup\lambda}$ corresponding to (10) only in the form $\displaystyle\binom{r_{\nu\mu}+\alpha_{P}(\mu_{l}+1)-1}{0}$ , which is equal to $\displaystyle\binom{r_{\nu\mu}+\alpha_{P}(\mu_{l}+1)}{0}$ .

∎

Lemma 27.

Let $\mu$ be as in Lemma 26. Let $\mu\subset\nu\subset R_{k+1-l}$ and assume $\nu/\mu$ is a horizontal strip. Let $0\leq x\leq\nu_{l}$ . Then we have

$(1)$ **

[TABLE]

$(2)$ * Let $P$ and $\alpha_{P}(u)$ $(u\in\mathbb{Z}_{>0})$ be as in Lemma 26 and assume $\max_{i}(t_{i})\leq\mu_{l}$ . Then*

[TABLE]

Proof.

$(1)$ : Since $\mu\subset R_{k+1-l}$ we have $\mathfrak{c}(\mu)=\mu$ . Since $\nu\subset R_{k+1-l}$ and $x\leq\nu_{l}$ , we have $\mathfrak{c}(\nu\cup(x))_{i}=(\nu\cup(x))_{i}$ for $i\neq 1$ . Thus $r_{\mathfrak{c}(\nu\cup(x)),\mathfrak{c}(\mu)}=r_{\nu\cup(x),\mu}$ . Moreover, $r_{\nu\cup(x),\mu}\neq r_{\nu,\mu}$ happens only if the $(l+1)$ -th part of $\nu\cup(x)$ blocks the $\mu$ -removable corner in the $l$ -th row, i.e. $x\geq\mu_{l}$ , in which case $r_{\nu\cup(x),\mu}=r_{\nu,\mu}-1$ .

$(2)$ :

Assume $t_{1}<\dots<t_{m}$ without loss of generality and thus $\max_{i}(t_{i})=t_{m}$ . We put $T=\mathfrak{c}(P)_{1}=\sum_{j}t_{j}$ . Since $\mathfrak{c}(P\cup\mu)=\mu\oplus\mathfrak{c}(P)$ , a removable corner $(r,c)$ of $\mathfrak{c}(P\cup\mu)$ satisfies one of the following:

•

(type 1) $r\geq l+1$ , and $(r-l,c)$ is a removable corner of $\mathfrak{c}(P)$ ,

•

(type 2) $c\geq T+1$ , and $(r,c-T)$ is a removable corner of $\mu$ .

We put

[TABLE]

for $j=1,2$ .

We denote by $i$ the residue of top addable corner of $\mathfrak{c}(P\cup\mu)$ . Then we have $X_{1}=\{i+t_{1},i+t_{2},\ldots,i+t_{m}\}$ and $i+\mu_{l}\in X_{2}\subset[i+\mu_{l},i+k-1]$ . Note that

[TABLE]

Next we show that, for $i\geq 1$ ,

[TABLE]

(11) is obvious since the smallest part of $\nu$ , which is $\nu_{l}$ , is greater than or equal to the largest part of $P\cup(x)$ , which is $\max\{x,t_{m}\}$ .

For (12), first we note that, by (11) and Figure 1 in the proof of Proposition 20, we have

[TABLE]

Then by Lemma 1 we have, for $1\leq i\leq l$ ,

[TABLE]

where we used $(P\cup\nu\cup(x))_{i}=\nu_{i}$ for $1\leq i\leq l$ for the first equality and $l+1\leq i+(k+1-\nu_{i})\leq l+(k+1-\mu_{n})$ (the first equality holds if and only if $i=1$ and $\nu_{1}=k+1-l$ ) for the second equality.

Thus we have $\mathfrak{c}(P\cup\nu\cup(x))_{i}=\mathfrak{c}(\nu\cup(x))_{i}+T$ for $1\leq i\leq l+1$ and $\mathfrak{c}(P\cup\nu\cup(x))_{i}\leq T$ for $i>l+1$ , which implies (12).

Hence, $|Y_{1}|=r_{\mathfrak{c}(P\cup(x)),\mathfrak{c}(P)}=\alpha_{P}(x+1)$ , and $|Y_{2}|=r_{\mathfrak{c}(\nu\cup(x)),\mu}=r_{\nu\mu}-\delta\left[x\geq\mu_{l}\right]$ .

Moreover $Y_{1}\cap Y_{2}=\{i+\mu_{l}\}$ if $x<t_{m}=\mu_{l}$ , and $Y_{1}\cap Y_{2}=\varnothing$ otherwise. Then

[TABLE]

∎

Thus Steps (A-1) and (B-1) have been achieved.

5.4. Steps (A-2) and (B-2)

The next lemma is technically important to perform the instructions in Step (A-2) and (B-2).

Lemma 28.

Let $\nu,u,p$ be as in the assumptions in Definition 25 and $n$ be an integer. Then we have the following equalities. In particular, in either case, the left-hand side does not depend on $p$ .

$(1)$ * $\displaystyle\sum_{i=0}^{u}\binom{p+n+i-1}{i}T_{\nu,u-i,p}=\sum_{s=0}^{u}\binom{n+s-1}{s}g^{(k)}_{\nu\cup(u-s)}$ .*

$(2)$ * $\displaystyle\sum_{i=0}^{u}\binom{p+n+i-1}{i}T^{\prime}_{P,\nu,u-i,p}=\sum_{s=0}^{u}\binom{n-\alpha_{P}(u+1-s)+s-1}{s}g^{(k)}_{P\cup\nu\cup(u-s)}$ .*

Proof.

Since both equality can be proved in a parallel manner, we prove (2) here. By the definition of $T^{\prime}_{P,\nu,u-i,p}$ , we have

[TABLE]

∎

Now we can express $g^{(k)}_{\lambda}$ (resp. $g^{(k)}_{P\cup\lambda}$ ) as a linear combination of $g^{(k)}_{\mu}h_{r}$ (resp. $g^{(k)}_{P\cup\mu}h_{r}$ ) as proposed in the description of Step (A-2) (resp. (B-2)).

Lemma 29.

Let $P$ and $\alpha_{P}(u)$ $(u\in\mathbb{Z}_{>0})$ be as in $(\mathsf{NP})$ . Let $\lambda,{\bar{\lambda}},{\bar{l}}$ be as in $\mathsf{(N\lambda)}$ in Section 5.1. Write $r=\lambda_{l(\lambda)}$ . Assume that ${\bar{l}}\geq 1$ and $\max_{i}(t_{i})\leq{\bar{\lambda}}_{\bar{l}}$ .

[TABLE]

Here, in all of the three expressions, the number $A_{\mu,\bar{\lambda},q}$ is defined by the following recursion formula:

[TABLE]

Notice that for each $\mu$ , $A_{\mu,\bar{\lambda},q}=0$ except for finitely many $q$ . The explicit value of $A_{\mu,\bar{\lambda},q}$ will be given in Lemma 34 below.

Remark. In the above recursion formula, $r_{\mu\mu}-r_{\mu\kappa}\geq 0$ always holds because there must be a $\mu$ -removable corner in every row in which there is a $\mu$ -nonblocked $\kappa$ -removable corner since $\mu/\kappa$ is a horizontal strip.

Proof.

(1) By Lemma 26(1),

[TABLE]

Then, by the definition of $A_{\nu,\bar{\lambda},q}$ , noting that $\bar{\lambda}\subsetneq\nu$ ,

[TABLE]

Hence,

[TABLE]

(3) is proved almost parallel to (1): By Lemma 26(2) and (3),

[TABLE]

Here the last equality follows from $\binom{-\alpha_{P}(r+1-s)+\alpha_{P}(r)-1+s}{s}=(-1)^{s}\binom{\alpha_{P}(r+1-s)-\alpha_{P}(r)}{s}$ and $0\leq\alpha_{P}(r+1-s)-\alpha_{P}(r)\leq s-1$ for $s\geq 1$ .

For (2), we have

[TABLE]

which is equal to $g^{(k)}_{P\cup\lambda}$ since this sum has exactly the same form as appeared in the proof of (3). ∎

In fact we can explicitly solve the recursion formula of $A_{\mu,\bar{\lambda},q}$ appeared in the previous proposition. This result is needed in the author’s following paper [Takigiku] and included in Appendix C.

Now Step (A) and (B) have been accomplished.

5.5. Step (C)

We multiply $g^{(k)}_{\lambda}$ by $g^{(k)}_{P}$ , and express it as a linear combination of $K$ - $k$ -Schur functions, and solve it:

Theorem 30.

Let $P$ and $\alpha_{P}(u)$ $($ for $u\in\mathbb{Z}_{>0})$ be as in $(\mathsf{NP})$ in Section 3, before Proposition 14. Let $\lambda,{\bar{\lambda}},{\bar{l}}$ be as in $\mathsf{(N\lambda)}$ in Section 5.1. Write $r=\lambda_{l(\lambda)}$ . Assume $\max_{i}\{t_{i}\}<{\bar{\lambda}}_{\bar{l}}$ . Then we have

[TABLE]

In particular, if $t_{n}<r$ then $\alpha_{P}(r)=0$ and

[TABLE]

Proof.

(2) follows from (1) and Lemma 21. We prove (1) by induction on ${\bar{l}}\geq 0$ . The case ${\bar{l}}=0$ was proved in Proposition 20 and Theorem 23. Assume ${\bar{l}}\geq 1$ .

From Lemma 29,

[TABLE]

∎

Now we can achieve our goal in this section.

Theorem 31.

For $1\leq t_{1}<\cdots<t_{m}\leq k$ and $a_{1},\dots,a_{m}>0$ ,

[TABLE]

Proof.

Use induction on $m>0$ .

The base case $m=1$ is obvious. Assume $m>1$ .

Applying Proposition 18 for $\lambda=R_{t_{m}}^{i}$ and $t=t_{m}$ , we have

[TABLE]

Multiplying this for $i=0,\dots,a_{m}-2$ , we have

[TABLE]

Put $P=R_{t_{1}}^{a_{1}}\cup\cdots\cup R_{t_{m-1}}^{a_{m-1}}$ .

Similarly applying Proposition 18 for $\lambda=P\cup R_{t_{m}}^{i}$ and $t=t_{m}$ , then multiplying this for $i=0,\dots,a_{m}-2$ , we have

[TABLE]

On the other hand, applying the previous theorem for $P$ , $\lambda=R_{t_{m}}$ , we have $g^{(k)}_{P\cup R_{t_{m}}}=g^{(k)}_{P}g^{(k)}_{R_{t_{m}}}$ .

Hence we have

[TABLE]

where the last equality follows by the induction hypothesis. ∎

6. Discussions

In this section we state some conjectures, that are consistent with our results in previous sections.

Conjecture 32.

For all $\lambda\in\mathcal{P}_{k}$ and $P=R_{t_{1}}^{a_{1}}\cup\dots\cup R_{t_{m}}^{a_{m}}$ , write

[TABLE]

Then $a_{P,\lambda,\mu}\geq 0$ for any $\mu$ .

In the case $P=R_{t}$ , it is observed that $a_{R_{t},\lambda,\mu}=0$ or $1$ . Moreover, the set of $\mu$ such that $a_{R_{t},\lambda,\mu}=1$ is expected to be an “interval”, but we have to consider the strong order on $\mathcal{P}_{k}\simeq\mathcal{C}_{k+1}\simeq\tilde{S}_{k+1}/S_{k+1}$ , which can be seen as just inclusion as shapes in the poset of cores, or strong Bruhat order on the affine symmetric group. Namely, the strong order $\lambda\leq\mu$ on $\mathcal{P}_{k}$ is defined by $\mathfrak{c}(\lambda)\subset\mathfrak{c}(\mu)$ . Notice that $\lambda\preceq\mu\Longrightarrow\lambda\subset\mu\Longrightarrow\lambda\leq\mu$ for $\lambda,\mu\in\mathcal{P}_{k}$ . Then,

Conjecture 33.

For all $\lambda\in\mathcal{P}_{k}$ and $1\leq t\leq k$ , there exists $\mu\in\mathcal{P}_{k}$ such that

[TABLE]

Assuming this conjecture, we shall write $\mathop{\mathrm{minindex}}(\lambda,t)=\mu$ .

We can make some conjectures about the behavior of minindex:

•

It is expected that if $\lambda$ gets “bigger” with respect to inclusion, then minindex gets weakly bigger in the strong order. Namely,

For any two elements $\mu\subset\lambda$ of $\mathcal{P}_{k}$ , we have $\mathop{\mathrm{minindex}}(\mu,t)\leq\mathop{\mathrm{minindex}}(\lambda,t)$ .

•

If a bounded partition has a form $R_{s}\cup\lambda$ for $s\neq t$ , its minindex still be expected to contains $R_{s}$ , and the “remaining part” is bigger or equal to $\mathop{\mathrm{minindex}}(\lambda,t)$ in the strong order:

For all $\lambda\in\mathcal{P}_{k}$ and $1\leq s\neq t\leq k$ , $\mathop{\mathrm{minindex}}(R_{s}\cup\lambda,t)$ has the form $R_{s}\cup\mu$ and $\mathop{\mathrm{minindex}}(\lambda,t)\leq\mu$ .

•

If a bounded partition has a form $R_{s}\cup R_{s}\cup\lambda$ for $s\neq t$ , its minindex would be equal to the union of $R_{s}$ and $\mathop{\mathrm{minindex}}(R_{s}\cup\lambda)$ :

For all $\lambda\in\mathcal{P}_{k}$ and $1\leq s\neq t\leq k$ , we have $R_{s}\cup\mathop{\mathrm{minindex}}(R_{s}\cup\lambda,t)=\mathop{\mathrm{minindex}}(R_{s}\cup R_{s}\cup\lambda,t)$ .

Next, consider a bounded partition that has a form $R_{t}\cup\lambda$ . We wrote

[TABLE]

On the other hand, by Proposition 18

[TABLE]

Now we can expect that for any $\mu\in\mathcal{P}_{k}$ ,

[TABLE]

where $I_{\mu,t}$ is an order filter of the interval $[\varnothing,R_{t}\cup\mu]$ (in $\mathcal{P}_{k}$ with the strong order) such that

[TABLE]

Appendix A Examples

In this section we sometimes abbreviate $\sum_{\lambda}a_{\lambda}g^{(3)}_{\lambda}$ as $\sum_{\lambda}a_{\lambda}\lambda$ for ease to see.

Appendix B Proof of Proposition 3

$(4)\Longrightarrow(1)$ **: The latter condition $\tau\prec\cdot\hskip 2.84526pt\exists\tau^{(1)}\prec\cdot\hskip 2.84526pt\dots\prec\cdot\hskip 2.84526pt\exists\tau^{(r)}=\kappa$ is obvious.

If $\kappa/\tau$ is not a horizontal strip, then $(a,b),(a+1,b)\in\kappa/\tau$ for $\exists a,b$ . Write their residues $i=b-a$ , $i-1=b-(a+1)$ . Then a $s_{i-1}$ -action should be performed after a $s_{i}$ -action.

Then the representation of (4) has the form $\kappa=\dots s_{i-1}\dots s_{i}\dots\tau$ , which contradicts (4).

$(1)\Longrightarrow(4)$ **: Assume $\kappa=\dots s_{i}\dots s_{i+1}\dots\tau$ , $|\kappa|_{k+1}=|\tau|_{k+1}+r$ , and $\kappa/\tau$ is a horizontal strip.

Consider the moment just before performing the action of $s_{i+1}$ . At that time the situation around each extremal cell of residue $i+1$ is one of the following:

*(1): **

$i+1$

** (2): **

$i+1$

** (3): **

$i+1$

** (4): **

$i+1$

*In the case (1), furthermore it should be **

$i+1$$i$

** since $\kappa/\tau$ is a horizontal strip. Besides, the case (1) should happen because the action of $s_{i+1}$ must add more than or equal to one box.*

In fact the case (4) never happens since the action of $s_{i+1}$ does not remove boxes.

The case (3) is divided to

*(3-1): **

$i+1$$i$

** and (3-2): **

$i+1$$i$

** .*

The case (3-1) should happen since later the action of $s_{i}$ must add more than or equal to one box.

Thus we have a contradiction that there are both addable corners and removable corners of residue $i$ in this moment.

Appendix C Explicit description of $A_{\mu,{\bar{\lambda}},q}$

Lemma 34.

In the setting of Lemma 29,

[TABLE]

Proof.

We fix $\bar{\lambda}$ , and set $f_{\mu}(t):=\sum_{q}A_{\mu,\bar{\lambda},q}t^{q}\in\mathbb{Z}[t]$ . Then the definition of $A_{\mu,\bar{\lambda},q}$ (in the statement of Lemma 29) is transformed into the recursion formula

[TABLE]

and the desired result becomes the condition

[TABLE]

We prove it by induction on $|\mu|$ (for $\mu$ satisfying $\bar{\lambda}\subset\mu\subset R^{\prime}_{{\bar{l}}}$ ). The base case $\mu=\bar{\lambda}$ is obvious by definition.

Then we assume $\bar{\lambda}\subsetneq\mu\subset R^{\prime}_{{\bar{l}}}$ . First we consider the case where $\mu/\bar{\lambda}$ is a vertical strip. In this case we put

[TABLE]

and we denote by $\kappa(c_{1},\ldots,c_{s})$ the partition defined by

[TABLE]

for $0\leq c_{i}\leq a_{i}$ ( $1\leq i\leq s$ ). In particular $\kappa(0,\ldots,0)=\bar{\lambda}$ and $\kappa(a_{1},\ldots,a_{s})=\mu$ .

Since $|\kappa(c_{1},\ldots,c_{s})/\bar{\lambda}|=\sum_{i}c_{i}$ and $r_{\kappa(c_{1},\ldots,c_{s})^{\prime}\bar{\lambda}^{\prime}}=r_{\bar{\lambda}\bar{\lambda}}-\#\{i\mid c_{i}=b_{i}\}$ , we have

[TABLE]

for $0\leq c_{i}\leq a_{i}$ and $(c_{1},\ldots,c_{s})\neq(a_{1},\ldots,a_{s})$ , by the induction hypothesis.

For $S\subset\{1,\ldots,s\}$ , we set $\kappa(S)=\kappa(a_{1}-\delta\left[1\in S\right],\ldots,a_{s}-\delta\left[s\in S\right])$ . Then

[TABLE]

Therefore,

[TABLE]

In fact it can be proved that $(X)=0$ by the following Claim 1 and Claim 2.

Claim 1.

[TABLE]

Proof of Claim 1: Reduced to next lemma:

Lemma 35.

Let $\gamma\subset\beta$ and $y=(r,c)$ be an addable corner of $\gamma$ . Put $\tilde{\gamma}=\gamma\cup\{y\}$ . Assume that $\tilde{\gamma}_{1}+l(\tilde{\gamma})\leq k+1$ and $y$ is $\beta$ -nonblocked. Then

[TABLE]

Proof of Lemma 35.

Note that $r_{\beta\tilde{\gamma}}=\#\{\text{$ \beta $-nonblocked$ \tilde{\gamma} $-removable corners}\}$ since $\tilde{\gamma}\subset R^{\prime}_{{\bar{l}}}$ , and the same equality holds for $r_{\beta\gamma}$ .

If $z$ is a $\beta$ -blocked (resp. nonblocked) removable corner of $\gamma$ other than $(r-1,c)$ or $(r,c-1)$ , then $z$ is also $\beta$ -blocked (resp. nonblocked) removable corner of $\tilde{\gamma}$ , and vice versa. Note that

•

$y=(r,c)$ is a $\beta$ -nonblocked removable corner of $\tilde{\gamma}$ , and not in $\gamma$ .

•

$(r,c-1)$ is not a removable corner of $\tilde{\gamma}$ .

•

$(r-1,c)$ is not a removable corner of $\tilde{\gamma}$ . Even if $(r-1,c)$ is a removable corner of $\gamma$ , it is $\beta$ -blocked.

Hence we conclude

[TABLE]

∎

Claim 2.

[TABLE]

for $\varnothing\neq T\subset\{2,\ldots,s\}$

Proof of Claim 2: Put $a^{\prime}_{i}=a_{i}-\delta\left[i\in T\right]$ , then

[TABLE]

(End of the proof of Claim 2)

Hence,

[TABLE]

This completes the proof in the case where $\mu/\bar{\lambda}$ is a vertical strip.

Next we consider the case where $\mu/\bar{\lambda}$ is not a vertical strip.

We take the same $x_{i}$ , $a_{i}$ , $b_{i}$ ( $1\leq i\leq s$ ) as above (in this case we have $a_{i}>b_{i}$ for some $i$ ), and $\kappa(c_{1},\ldots,c_{s})$ ( $0\leq c_{i}\leq a_{i},1\leq i\leq s$ , so long as adding $c_{i}$ cells on top of the $x_{i}$ -th column of ${\bar{\lambda}}$ , for all $i$ , yields a Young diagram of a partition) and $\kappa(S)$ ( $S\subset\{1,2,\dots,s\}$ but bound by the same restriction).

Notice that $\kappa(c_{1},\ldots,c_{s})/\bar{\lambda}$ is a vertical strip if and only if $c_{i}\leq b_{i}$ for all $i$ .

Now we have

[TABLE]

∎

Remark. By Lemma 34, Lemma 29(1), say, can be rewritten as:

[TABLE]

TL;DR

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Factorization formulas of KKK-kkk-Schur functions I

Abstract.

Contents

1. Introduction

2. Preliminaries

2.1. Partitions

2.2. Bounded partitions, cores, affine Grassmannian elements, and kkk-rectangles RtR_{t}Rt​

Lemma 1**.**

2.3. Weak order and weak strips

Definition-Proposition 2**.**

Proof.

Definition-Proposition 3**.**

Proof.

2.4. Symmetric functions

2.5. kkk-Schur functions

Definition 4** (kkk-Schur function via “weak Pieri rule”).**

Proposition 5** (kkk-rectangle property).**

2.6. KKK-kkk-Schur functions gλ(k)g^{(k)}_{\lambda}gλ(k)​

Definition 6** (affine set-valued strip).**

Definition 7** (KKK-kkk-Schur function via an “affine set-valued” Pieri rule).**

Definition 8**.**

Proposition 9**.**

2.7. Some properties of bounded partitions and cores

Corollary 10**.**

Proof.

Proposition 11**.**

Proof.

Corollary 12**.**

Proof.

Lemma 13**.**

Proof.

3. Possibility of factoring out gRt1a1∪⋯∪Rtmam(k)g^{(k)}_{R_{t_{1}}^{a_{1}}\cup\dots\cup R_{t_{m}}^{a_{m}}}gRt1​a1​​∪⋯∪Rtm​am​​(k)​ and some other general results

Proposition 14**.**

Proof.

Corollary 15**.**

Lemma 16**.**

Proof.

Corollary 17**.**

Proof.

Proposition 18**.**

Proof.

Theorem 19**.**

Proof.

4. A factorization of gRt1a1∪⋯∪Rtmam∪(r)(k)g^{(k)}_{R_{t_{1}}^{a_{1}}\cup\dots\cup R_{t_{m}}^{a_{m}}\cup(r)}gRt1​a1​​∪⋯∪Rtm​am​​∪(r)(k)​

Proposition 20**.**

Proof.

Lemma 21**.**

Proof.

Lemma 22**.**

Proof.

Theorem 23**.**

Proof.

5.1. Statements

5.2. Proofs

5.3. Steps (A-1) and (B-1)

Lemma 24**.**

Proof.

Definition 25**.**

Lemma 26**.**

Proof.

Lemma 27**.**

Proof.

5.4. Steps (A-2) and (B-2)

Lemma 28**.**

Proof.

Lemma 29**.**

Proof.

5.5. Step (C)

Theorem 30**.**

Proof.

Theorem 31**.**

Proof.

6. Discussions

Factorization formulas of $K$ - $k$ -Schur functions I

2.2. Bounded partitions, cores, affine Grassmannian elements, and $k$ -rectangles $R_{t}$

Lemma 1.

Definition-Proposition 2.

Definition-Proposition 3.

2.5. $k$ -Schur functions

Definition 4 ( $k$ -Schur function via “weak Pieri rule”).

Proposition 5 ( $k$ -rectangle property).

2.6. $K$ - $k$ -Schur functions $g^{(k)}_{\lambda}$

Definition 6 (affine set-valued strip).

Definition 7 ( $K$ - $k$ -Schur function via an “affine set-valued” Pieri rule).

Definition 8.

Proposition 9.

Corollary 10.

Proposition 11.

Corollary 12.

Lemma 13.

3. Possibility of factoring out $g^{(k)}_{R_{t_{1}}^{a_{1}}\cup\dots\cup R_{t_{m}}^{a_{m}}}$ and some other general results

Proposition 14.

Corollary 15.

Lemma 16.

Corollary 17.

Proposition 18.

Theorem 19.

4. A factorization of $g^{(k)}_{R_{t_{1}}^{a_{1}}\cup\dots\cup R_{t_{m}}^{a_{m}}\cup(r)}$

Proposition 20.

Lemma 21.

Lemma 22.

Theorem 23.

Lemma 24.

Definition 25.

Lemma 26.

Lemma 27.

Lemma 28.

Lemma 29.

Theorem 30.

Theorem 31.

Conjecture 32.

Conjecture 33.

Appendix C Explicit description of $A_{\mu,{\bar{\lambda}},q}$

Lemma 34.

Lemma 35.