Last update: 13 November 2012

In this section we shall explain how the definitions in Section 5.1 correspond to classical notions in Young tableaux theory. As in the previous section let $R$ be the root system of Type ${A}_{n-1}$ as given in Section 5.2. For clarity, we shall state all of the results in this section for placed shapes $(\gamma ,J)$ such that $\gamma $ is dominant and integral, i.e., $\gamma =({\gamma}_{1},\dots ,{\gamma}_{n})$ with ${\gamma}_{1}\le \dots \le {\gamma}_{n}$ and ${\gamma}_{i}\in \mathbb{Z}\text{.}$ This assumption is purely for notational clarity.

Let $(\gamma ,J)$ be a local region such that $\gamma $ is dominant and integral. Let ${w}_{p}\in {\mathcal{F}}^{(\gamma ,J)}$ and let $p$ be the corresponding standard tableau as defined by the map in Theorem 5.9. Then it follows from the definitions of $\gamma $ and ${w}_{p}$ in (5.6) and (5.4) that

$$\begin{array}{cc}\u27e8w\gamma ,{\epsilon}_{i}\u27e9=\u27e8\gamma ,{w}_{p}^{-1}{e}_{i}\u27e9=c\left({\text{box}}_{{w}_{p}^{-1}\left(i\right)}\right)=c\left(p\left(i\right)\right),& \text{(6.2)}\end{array}$$where $p\left(i\right)$ is the box of $p$ containing the entry $i\text{.}$

In classical standard tableau theory the *axial distance* between two boxes in a standard tableau is defined as follows. Let $\lambda $ be a
partition and let $p$ be a standard tableau of shape $\lambda \text{.}$ Let
$1\le i,j\le n$ and let
$p\left(i\right)$ and
$p\left(j\right)$ be the boxes which are filled with $i$ and
$j,$ respectively. Let
$({r}_{i},{c}_{i})$ and
$({r}_{j},{c}_{j})$ be the positions of these boxes, where the
rows and columns of $\lambda $ are numbered in the same way as for matrices. Then the axial distance from $j$ to
$i$ in $p$ is

(see [Wen1988]). Rewriting this in terms of the local region $(\gamma ,J)$ determined by (5.7),

$${d}_{ji}\left(p\right)=c\left(p\left(j\right)\right)-c\left(p\left(i\right)\right)=\u27e8{w}_{p}\gamma ,{\epsilon}_{j}-{\epsilon}_{i}\u27e9={d}_{{\epsilon}_{j}-{\epsilon}_{i}}\left(w\right),$$where ${w}_{p}\in {\mathcal{F}}^{(\gamma ,J)}$ is the permutation corresponding to the standard tableau $p$ and ${d}_{\alpha}\left({w}_{p}\right)$is the $\alpha \text{-axial}$ distance defined in (f) of Section 5.1. This shows that the axial distance defined in (f) of Section 5.1 is a generalization of the classical notion of axial distance. These numbers are crucial to the classical construction of the seminormal representations of the symmetric group given by Young (see Remark (3) of Section 5.1).

The following proposition shows that, in the case of a root system of type A, the definition of skew local region coincides with the classical notion of a skew shape.

**Proposition 6.4.** Let $(\gamma ,J)$ be a local region
with $\gamma $ dominant and integral. Then the configuration of boxes associated to
$(\gamma ,J)$ is a placed skew shape if and only if
$(\gamma ,J)$ is a skew local region.

Proof. | |

$(\Leftarrow )$ We shall show that if the placed configuration corresponding to the pair $(\gamma ,J)$ has any $2\times 2$ blocks of the forms $$\begin{array}{ccc}\n\n\n\n\n\n\na\nb\nc\n\n& \n\n\n\n\n\n\na\nb\nc\n\n& \n\n\n\n\n\n\na\nb\n\n\\ \text{Case (1)}& \text{Case (2)}& \text{Case (3)}\end{array}$$then there exists a $w\in {\mathcal{F}}^{(\gamma ,J)}$ such that $w\gamma $ violates one of the two conditions in (c) of Section 5.1. This will show that if $(\gamma ,J)$ is a skew local region then the corresponding placed configuration of boxes must be a placed skew shape. In the pictures above the shaded regions indicate the absence of a box and, for reference, we have labeled the boxes with $a,b,c\text{.}$
by filling the region of the configuration strictly north and weakly west of box c in row reading order (sequentially left to right across the rows starting at the top), putting the next entry in box c, and filling the remainder of the configuration in column reading order (sequentially down the columns beginning at the leftmost available column). Let $w={w}_{p}$ be the permutation in ${\mathcal{F}}^{(\gamma ,J)}$ which corresponds to the standard tableau $p\text{.}$ Let $p\left(i\right)$ denote the box containing $i$ in $p\text{.}$ Then, using the identity (6.2), $$\u27e8w\gamma ,{\alpha}_{i}+{\alpha}_{i+1}\u27e9=\u27e8w\gamma ,{\epsilon}_{i+1}-{\epsilon}_{i-1}\u27e9=c\left(p(i+1)\right)-c\left(p(i-1)\right)=0,$$since the boxes $p(i+1)$ and $p(i-1)$ are on the same diagonal. However, $$\begin{array}{ccc}\u27e8w\gamma ,{\alpha}_{i}\u27e9& =& \u27e8w\gamma ,{e}_{i}-{\epsilon}_{i-1}\u27e9=c\left(p\left(i\right)\right)-c\left(p(i-1)\right)=1,\phantom{\rule{1em}{0ex}}\text{and}\\ \u27e8w\gamma ,{\alpha}_{i+1}\u27e9& =& \u27e8w\gamma ,{\epsilon}_{i+1}-{\epsilon}_{i}\u27e9=c\left(p(i+1)\right)-c\left(p(i)\right)=-1,\end{array}$$and so condition (c)(1) of Section 5.1 is violated.
by filling the region weakly north and strictly west of box c in column reading order, putting the next entry in box c, and filling the remainder of the configuration in row reading order. Using this standard tableau $p,$ the remainder of the argument is the same as for Case (1).
by filling the region strictly north and strictly west of box b in column reading order, putting the next entry in box b, and filling the remainder of the configuration in row reading order. Let $w={w}_{p}$ be the permutation in ${\mathcal{F}}^{(\gamma ,J)}$ corresponding to $p$ and let $p\left(i\right)$ denote the box containing $i$ in $p\text{.}$ Then $$\u27e8w\gamma ,{\alpha}_{i}\u27e9=\u27e8w\gamma ,{\epsilon}_{i}-{\epsilon}_{i-1}\u27e9=c\left(p\left(i\right)\right)-\left(p(i-1)\right)=0,$$since $t\left(i\right)$ and $t(i-1)$ are on the same diagonal. Hence, condition (c)(1) of Section 5.1 is violated. $(\Rightarrow )$ Let $\gamma \in {\mathbb{Z}}^{n}$ and $\lambda /\mu $ describe a placed skew shape (a skew shape placed on infinite graph paper). Let $(\gamma ,J)$ be the corresponding local region as defined in (5.7). We will show that every $w\gamma $ is calibratable for every $w\in {\mathcal{F}}^{(\gamma ,J)}\text{.}$ Let $w\in {\mathcal{F}}^{(\gamma ,J)}$ and let $p$ be the corresponding standard tableau of shape $\lambda /\mu \text{.}$ Consider a $2\times 2$ block of boxes of $p\text{.}$ If these boxes are filled with $$\n\n\n\n\n\n\ni\nj\nk\n\u2113\n\n$$then either $i<j<k<\ell $ or $i<k<j<\ell \text{.}$ In both cases we have $i<\ell -1$ and it follows that $\ell -1$ and $\ell $ are not on the same diagonal. Thus $$\u27e8w\gamma ,{\alpha}_{\ell}\u27e9=c\left(p\left(\ell \right)\right)-c\left(p(\ell -1)\right)\ne 0,$$and so $w\gamma $ satisfies condition (a) in the definition of calibratable. The same argument shows that one can never get a standard tableau in which $\ell $ and $\ell -2$ occur in adjacent boxes of the same diagonal and thus it follows that $w\gamma $ satisfies condition (b) in the definition of calibratable. Thus $(\gamma ,J)$ is a skew local region. $\square $ |

Classically, a *border strip* (or *ribbon*) is a skew shape which contains at most one box in each diagonal. Although the convention,
[Mac1995, I, Section 1 p. 5], is to assume that border strips are connected skew shapes we shall *not* assume this.

Recall from (b) of Section 5.1 that a placed shape $(\gamma ,J)$ is a placed ribbon shape if $\gamma $ is regular, i.e., $\u27e8\gamma ,\alpha \u27e9\ne 0$ for all $\alpha \in R\text{.}$

**Proposition 6.6.** Let $(\gamma ,J)$ be a placed ribbon
shape such that $\gamma $ is dominant and integral. Then the configuration of boxes corresponding to
$(\gamma ,J)$ is a placed border strip.

Proof. | |

Let $(\gamma ,J)$ be a placed ribbon shape with $\gamma $ dominant and regular. Since $\gamma =({\gamma}_{1},\dots ,{\gamma}_{n})$ is regular, ${\gamma}_{i}\ne {\gamma}_{j}$ for all $i\ne j\text{.}$ In terms of the placed configuration ${\gamma}_{i}=c\left({\text{box}}_{i}\right)$ is the diagonal that ${\text{box}}_{i}$ is on. Thus the configuration of boxes corresponding to $(\gamma ,J)$ contains at most one box in each diagonal. $\square $ |

**Example.** If
$\gamma =(-6,-5,-4,0,1,3,4,5,6,7)$
and
$J=\{{\epsilon}_{2}-{\epsilon}_{1},{\epsilon}_{5}-{\epsilon}_{4},{\epsilon}_{7}-{\epsilon}_{6},{\epsilon}_{9}-{\epsilon}_{8},{\epsilon}_{10}-{\epsilon}_{9}\}$
then the placed configuration of boxes corresponding to $(\gamma ,J)$ is the placed border strip

where the boxes are labeled with their contents.

Let $(\gamma ,J)$ be a placed shape with $\gamma $ dominant and integral (i.e., $\gamma =({\gamma}_{1},\dots ,{\gamma}_{n})$ with ${\gamma}_{1}\le \dots \le {\gamma}_{n}$ and ${\gamma}_{i}\in \mathbb{Z}\text{)}$ and view $(\gamma ,J)$ as a placed configuration of boxes. In terms of placed configurations, conjugation of shapes is equivalent to transposing the placed configuration across the diagonal of boxes of content 0. The following example illustrates this.

**Example.** Suppose
$\gamma =(-1,-1,-1,0,0,1,1)$
and
$J=({\epsilon}_{4}-{\epsilon}_{2},{\epsilon}_{4}-{\epsilon}_{3},{\epsilon}_{6}-{\epsilon}_{5},{\epsilon}_{7}-{\epsilon}_{5})\text{.}$
Then the placed configuration of boxes corresponding to $(\gamma ,J)$ is

in which the shaded box is not a box in the configuration.

The minimal length representative of the coset ${w}_{0}{W}_{\gamma}$ is the permutation

$$u=\left(\begin{array}{ccccccc}1& 2& 3& 4& 5& 6& 7\\ 5& 6& 7& 3& 4& 1& 2\end{array}\right)\text{.}$$We have $-u\gamma =-{w}_{0}\gamma =(-1,-1,0,0,1,1,1)$ and

$$\begin{array}{ccc}-u(p\left(\gamma \right)\backslash J)& =& -u\{{\epsilon}_{4}-{\epsilon}_{1},{\epsilon}_{5}-{\epsilon}_{1},{\epsilon}_{5}-{\epsilon}_{2},{\epsilon}_{5}-{\epsilon}_{3},{\epsilon}_{6}-{\epsilon}_{4},{\epsilon}_{7}-{\epsilon}_{4}\}\\ & =& -\{{\epsilon}_{3}-{\epsilon}_{5},{\epsilon}_{4}-{\epsilon}_{5},{\epsilon}_{4}-{\epsilon}_{6},{\epsilon}_{4}-{\epsilon}_{7},{\epsilon}_{1}-{\epsilon}_{3},{\epsilon}_{2}-{\epsilon}_{3}\}\\ & =& \{{\epsilon}_{5}-{\epsilon}_{3},{\epsilon}_{5}-{\epsilon}_{4},{\epsilon}_{6}-{\epsilon}_{4},{\epsilon}_{7}-{\epsilon}_{4},{\epsilon}_{3}-{\epsilon}_{1},{\epsilon}_{3}-{\epsilon}_{2}\}\text{.}\end{array}$$Thus the configuration of boxes corresponding to the placed shape ${(\gamma ,J)}^{\prime}$ is

$$\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n1\n\n0\n1\n\n-1\n1\n\n-1\n0\n\n\n$$
Let $(\gamma ,J)$ be a placed shape such that $\gamma $ is dominant and
integral and consider the placed configuration of boxes corresponding to
$(\gamma ,J)\text{.}$ The *minimal box* of the configuration is the
box such that

- $\left({m}_{1}\right)$ there is no box immediately above,
- $\left({m}_{2}\right)$ there is no box immediately to the left,
- $\left({m}_{3}\right)$ there is no box northwest in the same diagonal, and
- $\left({m}_{4}\right)$ it has the minimal content of the boxes satisfying $\left({m}_{1}\right)-\left({m}_{3}\right)\text{.}$

There is at most one box in each diagonal satisfying $\left({m}_{1}\right)-\left({m}_{3}\right)\text{.}$ Thus, $\left({m}_{4}\right)$ guarantees that the minimal box is unique. It is clear that the minimal box of the configuration always exists.

The *column reading* tableaux of shape $(\gamma ,J)$ is the filling
${p}_{\text{min}}$ which is created inductively by

- filling the minimal box of the configuration with 1, and
- if $1,2,\dots ,i$ have been filled in then fill the minimal box of the configuration formed by the unfilled boxes with $i+1\text{.}$

The *row reading tableau* of shape $(\gamma ,J)$ is the standard tableau
${p}_{\text{max}}$ whose conjugate ${p}_{\text{max}}^{\prime}$ is the column
reading tableaux for the shape ${(\gamma ,J)}^{\prime}$ (the conjugate shape to
$(\gamma ,J)\text{).}$

Recall the definitions of the weak Bruhat order and closed subsets of roots given after Eq. (4.5).

**Theorem 6.9.** Let $(\gamma ,J)$ be a placed shape such
that $\gamma $ is dominant and integral (i.e.,
$\gamma =({\gamma}_{1},\dots ,{\gamma}_{n})$
with ${\gamma}_{1}\le \dots \le {\gamma}_{n}$ and
${\gamma}_{i}\in \mathbb{Z}\text{).}$ Let
${p}_{\text{min}}$ and ${p}_{\text{max}}$ be the column reading
and row reading tableaux of shape $(\gamma ,J),$ respectively, and let
${w}_{\text{min}}$ and ${w}_{\text{max}}$ be the corresponding permutations in
${\mathcal{F}}^{(\gamma ,J)}\text{.}$ Then

where ${K}^{c}$ denotes the complement of $K$ in ${R}^{+}$ and

Proof. | |

(a) Consider the configuration of boxes corresponding to $(\gamma ,J)\text{.}$ If $k>i$ then either $c\left({\text{box}}_{k}\right)>c\left({\text{box}}_{i}\right),$ or ${\text{box}}_{k}$ is in the same diagonal and southeast of ${\text{box}}_{i}\text{.}$ Thus when we create ${p}_{\text{min}}$ we have that $$\text{if}\phantom{\rule{1em}{0ex}}k>i\phantom{\rule{1em}{0ex}}\text{then}\phantom{\rule{1em}{0ex}}{\text{box}}_{k}\phantom{\rule{0.2em}{0ex}}\text{gets filled before}\phantom{\rule{0.2em}{0ex}}{\text{box}}_{i}\iff {\text{box}}_{k}\phantom{\rule{0.2em}{0ex}}\text{is}\phantom{\rule{0.2em}{0ex}}\text{northwest}\phantom{\rule{0.2em}{0ex}}\text{of}\phantom{\rule{0.2em}{0ex}}{\text{box}}_{i},$$
where the such that ${\text{box}}_{{i}_{m}}$ is either directly above ${\text{box}}_{{i}_{m-1}}$ or in the same diagonal and directly northwest of ${\text{box}}_{{i}_{m-1}}\text{.}$ In other words, $$\text{if}\phantom{\rule{1em}{0ex}}k>i\phantom{\rule{1em}{0ex}}\text{then}\phantom{\rule{1em}{0ex}}{p}_{\text{min}}\left({\text{box}}_{k}\right)<{p}_{\text{min}}\left({\text{box}}_{i}\right)\iff {\text{box}}_{k}\phantom{\rule{0.2em}{0ex}}\text{is}\phantom{\rule{0.2em}{0ex}}\text{northwest}\phantom{\rule{0.2em}{0ex}}\text{of}\phantom{\rule{0.2em}{0ex}}{\text{box}}_{i}\text{.}$$So, from the formula for ${w}_{p}$ in (5.4) we get $$\text{if}\phantom{\rule{1em}{0ex}}k>i\phantom{\rule{1em}{0ex}}\text{then}\phantom{\rule{1em}{0ex}}{w}_{\text{min}}\left(k\right)<{w}_{\text{min}}\left(i\right)\iff {\epsilon}_{k}-{\epsilon}_{i}\in \stackrel{\u203e}{J},$$where ${w}_{\text{min}}$ is the permutation in ${\mathcal{F}}^{(\gamma ,J)}$ which corresponds to the filling ${t}_{\text{min}}$ and $\stackrel{\u203e}{J}$ is the closure of $J$ in $R\text{.}$ It follows that $$R\left({w}_{\text{min}}\right)=\stackrel{\u203e}{J}\text{.}$$(b) There are at least two ways to prove that $R\left({w}_{\text{max}}\right)=\stackrel{\u203e}{(P\left(\gamma \right)\backslash J)\cup Z{\left(\gamma \right)}^{c}}\text{.}$ One can mimic the proof of part (a) by defining the maximal box of a configuration and a corresponding filling. Alternatively one can use the definition of conjugation and the fact that $R\left({w}_{0}w\right)=R{\left(w\right)}^{c}\text{.}$ The permutation ${w}_{\text{min}}$ is the unique minimal element of ${\mathcal{F}}^{(\gamma ,J)}$ and the conjugate of ${w}_{\text{max}}$ is the unique minimal element of ${\mathcal{F}}^{{(\gamma ,J)}^{\prime}}\text{.}$ We shall leave the details to the reader. (c) An element $w\in W$ is an element of ${\mathcal{F}}^{(\gamma ,J)}$ if and only if $R\left(w\right)\cap P\left(\gamma \right)=J$ and $R\left(w\right)\cap Z\left(\gamma \right)=\varnothing \text{.}$ Thus ${\mathcal{F}}^{(\gamma ,J)}$ consists of those permutations $w\in W$ such that $$\stackrel{\u203e}{J}\subseteq R\left(w\right)\subseteq \stackrel{\u203e}{(P\left(\gamma \right)\backslash J)\cup Z{\left(\gamma \right)}^{c}}\text{.}$$Since the weak Bruhat order is the ordering determined by inclusions of $R\left(w\right),$ it follows that ${\mathcal{F}}^{(\gamma ,J)}$ is the interval between ${w}_{\text{min}}$ and ${w}_{\text{max}}\text{.}$ $\square $ |

**Example.** Suppose
$\gamma =(-1,-1,-1,0,0,1,1)$
and
$J=\{{\epsilon}_{4}-{\epsilon}_{2},{\epsilon}_{4}-{\epsilon}_{3},{\epsilon}_{6}-{\epsilon}_{5},{\epsilon}_{7}-{\epsilon}_{5}\}\text{.}$
The minimal and maximal elements in ${\mathcal{F}}^{(\gamma ,J)}$ are the
permutations

The permutations correspond to the standard tableaux

$$\begin{array}{ccc}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n1\n2\n5\n\n3\n6\n\n4\n7\n\n\n& \text{and}& \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n1\n2\n3\n\n5\n4\n\n6\n7\n\n\n\end{array}$$
This is an excerpt of the paper entitled *Affine Hecke algebras and generalized standard Young tableaux* written by Arun Ram in 2002, published in the
Academic Press Journal of Algebra **260** (2003) 367-415. The paper was dedicated to Robert Steinberg.

Research supported in part by the National Science Foundation (DMS-0097977) and the National Security Agency (MDA904-01-1-0032).