## Standard tableaux for type C in terms of boxes

Last update: 15 November 2012

## Standard tableaux for type C in terms of boxes

### 8.1. The root system

Let $\left\{{\epsilon }_{1},\dots ,{\epsilon }_{n}\right\}$ be an orthonormal basis of ${𝔥}_{ℝ}^{*}={ℝ}^{n}$ and view elements $\gamma =\sum _{i}{\gamma }_{i}{\epsilon }_{i}$ of ${ℝ}^{n}$ as sequences

$γ= ( γ-n,…, γ-1;γ1 ,…,γn ) ,such that γ-i= -γi. (8.2)$

The root system of type ${C}_{n}$ is given by the sets

$R = { ±2εi, ±(εj±εi) ∣1≤i, j≤n } and R+ = { 2εi,εj ±εi∣ 1≤i

The simple roots are given by ${a}_{1}=2{\epsilon }_{1},$ ${\alpha }_{i}={\epsilon }_{i}-{\epsilon }_{i-1},$ $1\le i\le n\text{.}$ The Weyl group $W=W{C}_{n}$ is the hyperoctahedral group of permutations of $-n,\dots ,-1,1,\dots ,n$ such that $w\left(-i\right)=-w\left(i\right)\text{.}$ This groups acts on the ${\epsilon }_{i}$ by the rule $w{\epsilon }_{i}={\epsilon }_{w\left(i\right)},$ with the convention that ${\epsilon }_{-i}=-{\epsilon }_{i}\text{.}$

For this type C case there is a nice trick. View the root system as

$R = { ±(εj±εi) ∣1

with the convention that ${\epsilon }_{-i}=-{\epsilon }_{i}\text{.}$ In this notation ${\epsilon }_{i}-{\epsilon }_{-i}=2{\epsilon }_{i}$ and ${\epsilon }_{-i}-{\epsilon }_{-j}={\epsilon }_{j}-{\epsilon }_{i}\text{.}$ This way the type C root system “looks like” a type A root system and many computations can be done in the same way as in type A.

### 8.5. Rearranging $\gamma$

We analyze the structure of the sets ${ℱ}^{\left(\gamma ,J\right)}$ as considered in (4.3). This corresponds to when the $q$ in the affine Hecke algebra is not a root of unity. The analysis in this case is analogous to the method that was used in Section 5.11 to create books of placed configurations in the type A case.

Let $\gamma \in {ℝ}^{n}\text{.}$ Apply an element of the Weyl group to $\gamma$ to “arrange” the entries of $\gamma$ so that, for each $i\in \left\{1,\dots ,n\right\},$

$γi∈ [z+12,z], for somez∈ℤ.$

Then

$γ-i=-γi∈ [z′,z′+12] .for somez′∈ ℤ.$

As in the type A case, the sets $Z\left(\gamma \right)$ and $P\left(\gamma \right)$ can be partitioned according to the $ℤ$ cosets of the elements of $\gamma$ and it is sufficient to consider each $ℤ\text{-coset}$ separately and then assemble the results in “books of pages.” There are three cases to consider:

Case $β.$ The $ℤ\text{-coset}$ $\beta +ℤ,$ $\beta \in \left(1/2,1\right)\text{.}$ Then

$γ= ( -β-zn≤…≤-β -z2≤-β-z1; β+z1≤β+z2 ≤…≤β+zn ) ,z1∈ℤ,$

Case 1/2. The $ℤ\text{-coset}$ $1/2+ℤ\text{.}$ Then

$γ = ( -1/2-zn≤… ≤-1/2-z2≤- 1/2-z1; 1/2+z1≤1/2 +z2≤…≤1/2 +zn ) ,zi∈ ℤ≥0,$

Case 0. The $ℤ\text{-coset}$ $ℤ\text{.}$ Then

$γ= ( -zn≤…≤-z2 ≤-z1;z1 ≤z2≤…≤zn ) ,zi∈ℤ≥0.$

It is notationally convenient to let ${z}_{-i}=-{z}_{i}\text{.}$

### 8.6. Boxes and standard tableaux

Let us assume that the entries of $\gamma$ all lie in a single $ℤ\text{-coset}$ and describe the resulting standard tableaux. The general case is obtained by creating books of pages of standard tableaux where the pages correspond to the different $ℤ\text{-cosets}$ of entries in $\gamma \text{.}$

The placed configuration of boxes is determined as follows.

### 8.7. Case $\beta ,\beta \in \left(1/2,1\right)$

Assume that $\gamma \in {𝔥}_{ℝ}^{*}$ is of the form

$γ= ( -β-zn≤…≤-β- z2≤-β-z1; β+z1≤β+z2≤ …≤β+zn ) ,zi∈ℤ.$

Place boxes on two pages of infinite graph paper. These pages are numbered $\beta$ and $-\beta$ and each page has the diagonals numbered consecutively with the elements of $ℤ,$ from bottom left to top right. View these two pages, page $\beta$ and page $-\beta ,$ as “linked.” For each $1\le i\le n$ place ${\text{box}}_{i}$ on diagonal ${z}_{i}$ of page $\beta$ and ${\text{box}}_{-i}$ on diagonal $-{z}_{i}$ of page $-\beta \text{.}$ The boxes on each diagonal are arranged in increasing order from top left to bottom right. The placement of boxes on page $-\beta$ is a ${180}^{\circ }$ rotation of the placement of the boxes on page $\beta \text{.}$

Using the notation for the root system of type ${C}_{n}$ in (8.4)

$P(γ) = { εj-εi∣ j>iand boxiandboxj are in adjacent diagonals } and Z(γ) = { εj-εi∣ j>iand boxiandboxj are in the same diagonal } .$

Note that ${\epsilon }_{-i}-{\epsilon }_{-j}\in Z\left(\gamma \right)$ if and only if ${\epsilon }_{j}-{\epsilon }_{i}\in {Z}_{\beta }\left(\gamma \right),$ and similarly ${\epsilon }_{-i}-{\epsilon }_{-j}\in {P}_{\beta }\left(\gamma \right)$ if and only if ${\epsilon }_{j}-{\epsilon }_{i}\in {P}_{\beta }\left(\gamma \right)\text{.}$ If $J\subseteq P\left(\gamma \right)$ arrange the boxes on adjacent diagonals according to the rules

1. if ${\epsilon }_{j}-{\epsilon }_{i}\in J$ place ${\text{box}}_{j}$ northwest of ${\text{box}}_{i},$ and
2. if ${\epsilon }_{j}-{\epsilon }_{i}\in P\left(\gamma \right)\J$ place ${\text{box}}_{j}$ southeast of ${\text{box}}_{i}\text{.}$

A standard tableau is a negative rotationally symmetric filling $p$ of the $2n$ boxes with $-n,\dots ,-1,1,\dots ,n$ such that

1. $p\left({\text{box}}_{i}\right) if $j>i$ and ${\text{box}}_{j}$ and ${\text{box}}_{i}$ are in the same diagonal,
2. $p\left({\text{box}}_{i}\right)>p\left({\text{box}}_{j}\right)$ if $j>i,$ ${\text{box}}_{i}$ and ${\text{box}}_{j}$ are in adjacent diagonals and ${\text{box}}_{j}$ is northwest of ${\text{box}}_{i},$
3. $p\left({\text{box}}_{i}\right) if $j>i,$ ${\text{box}}_{i}$ and ${\text{box}}_{j}$ are in adjacent diagonals and ${\text{box}}_{j}$ is southeast of ${\text{box}}_{i}\text{.}$

The negative rotational symmetry means that the filling of the boxes on page $-\beta$ is the same as the filling on page $\beta$ except rotated by ${180}^{\circ }$ and with all entries in the boxes multiplied by $-1\text{.}$

Example. Suppose $\beta \in \left(1/2,1\right),$ and

$γ = (-β;β) + ( -2,-2,-2, -1,-1,-1, 0,0,0, 1,1,1; -1,-1,-1, 0,0,0, 1,1,1, 2,2,2 ) = ( -β-2, -β-2, -β-2, -β-1, -β-1, -β-1, -β,-β,-β, -β+1, -β+1, -β+1; β-1, β-1, β-1, β,β,β, β+1, β+1, β+1, β+2, β+2, β+2$

and

$J = { ε4-ε1, ε-1-ε-4, ε4-ε2, ε-2-ε-1, ε4-ε3, ε-3-ε-4, ε5-ε2, ε-2-ε-5, ε5-ε3, ε-3-ε-5, ε7-ε5, ε-5-ε-7, ε7-ε6, ε-6-ε-7, ε8-ε6, ε-6-ε-8, ε10-ε9, ε-9-ε-10, ε10-ε8, ε-8-ε-10, ε10-ε7, ε-7-ε-10, ε11-ε9, ε-9-ε-11, ε11-ε8, ε-8-ε-11, ε11-ε7, ε-7-ε-11, ε12-ε9 ε-9-ε-12 } .$

The placed configuration of boxes corresponding to $\left(\gamma ,J\right)$ is

$-1 0 1 2 -2 -1 1 2 0 1 0 1 -1 0 -1 0 -2 -1 1 2 -2 -1 0 1 Page-β Pageβ -9 -6 -3 10 -12 -8 -2 11 -5 -1 4 7 -7 -4 1 5 -11 2 8 12 -10 3 6 9 Page-β Pageβ contents of boxes numbering of boxes$

and a sample negative rotationally symmetric standard tableau is

$-4 -3 5 -11 -6 -1 7 -10 8 9 -9 -8 2 12 -9 -8 10 -7 1 6 11 -5 3 4 Page-β Pageβ a standard tableau$

### 8.8. Case 1/2

Assume that $\gamma \in {𝔥}_{ℝ}^{*}$ is of the form

$γ = ( -1/2-zn≤…≤ -1/2-z2≤-1/ 2-z1; 1/2+z1≤1/2+ z2≤…≤1/2+ zn ) ,zi∈ ℤ≥0.$

Place boxes on a page of infinite graph paper which has its diagonals numbered consecutively with the elements of $1/2+ℤ,$ from bottom left to top right. This page has page number $1/2\text{.}$ For each $i\in \left\{±1,\dots ,±n\right\}$ place boxi on diagonal $1/2+{z}_{i}$ and ${\text{box}}_{-i}$ on diagonal $-1/2-{z}_{i}\text{.}$ The boxes on each diagonal are arranged in increasing order from top left to bottom right and the placement of boxes is negative rotationally symmetric in the sense that a ${180}^{\circ }$ rotation takes boxi to ${\text{box}}_{-i}\text{.}$

Using the root system notation in (8.4),

$P(γ) = { εj-εi∣ j>iand boxiandboxj are in adjacent diagonals } and Z(γ) = { εj-εi∣ j>iand boxiandboxj are in the same diagonal } .$

Note that it is the formulation of the root system of type ${C}_{n}$ in (8.4) which makes the description of $P\left(\gamma \right)$ and $Z\left(\gamma \right)$ nice in this case. If $J\subseteq P\left(\gamma \right)$ arrange the boxes on adjacent diagonals according to the rules

1. if ${\epsilon }_{j}-{\epsilon }_{i}\in J$ place ${\text{box}}_{j}$ northwest of ${\text{box}}_{i},$ and
2. if ${\epsilon }_{j}-{\epsilon }_{i}\in P\left(\gamma \right)\J$ place ${\text{box}}_{j}$ southeast of ${\text{box}}_{i}\text{.}$

A standard tableau is a negative rotationally symmetric filling $p$ of the $2n$ boxes with $-n,\dots ,-1,1,\dots ,n$ such that

1. $p\left({\text{box}}_{i}\right) if $j>i$ and ${\text{box}}_{j}$ and ${\text{box}}_{i}$ are in the same diagonal,
2. $p\left({\text{box}}_{i}\right)>p\left({\text{box}}_{j}\right)$ if $j>i,$ ${\text{box}}_{i}$ and ${\text{box}}_{j}$ are in adjacent diagonals and ${\text{box}}_{j}$ is northwest of ${\text{box}}_{i},$
3. $p\left({\text{box}}_{i}\right) if $j>i,$ ${\text{box}}_{i}$ and ${\text{box}}_{j}$ are in adjacent diagonals and ${\text{box}}_{j}$ is southeast of ${\text{box}}_{i}\text{.}$

The negative rotational symmetry means that the filling of the boxes is the same if each entry is multiplied by $-1$ and the configuration is rotated by ${180}^{\circ }\text{.}$

Example. Suppose

$γ= ( -72, -52, -52, -32, -32, -32, -32, -12, -12, -12, -12, 12, 12, 12, 12, 32, 32, 32, 32, 52, 52, 72 )$

and

$J = { ε11-ε10, ε-10-ε-11, ε10-ε8, ε-8-ε-10, ε9-ε7, ε-7-ε-9, ε9-ε8, ε-8-ε-9, ε7-ε3, ε-3-ε-7, ε7-ε4, ε-4-ε-7, ε6-ε2, ε-2-ε-6, ε6-ε3, ε-3-ε-6, ε6-ε4, ε-4-ε-6, ε5-ε4, ε-4-ε-5, ε5-ε3, ε-3-ε-5, ε5-ε2, ε-2-ε-5, ε2-ε-1, ε3-ε-1, ε4-ε-1, ε1-ε-1 } . = { ε11-ε10, ε10-ε8, ε9-ε7, ε9-ε8, ε7-ε3, ε7-ε4, ε6-ε2, ε6-ε3, ε6-ε4, ε5-ε4, ε5-ε3, ε5-ε2, ε2+ε1, ε3+ε1, ε4+ε1, 2ε1 } .$

The placed configuration of boxes corresponding to $\left(\gamma ,J\right)$ is as given below:

$12 -32 -12 32 -52 -12 32 52 -72 -32 -12 12 32 72 -52 -32 12 52 -32 12 32 -12 1 -8 -4 5 -10 -3 6 9 -11 -7 -2 2 7 11 -9 -6 3 10 -5 4 8 -1 -11 -7 -6 -10 -5 -4 -9 -8 -3 -2 -1 1 2 3 8 9 4 5 10 6 7 11 Page12 Page12 Page12 contents of boxes numbering of boxes a standard tableau$

### 8.9. Case 0

Assume that $\gamma \in {𝔥}_{ℝ}^{*}$ is of the form

$γ= ( -zn≤…≤-z2 ≤-z1; z1≤z2≤…≤ zn ) ,zi∈ℤ≥0.$

Place boxes on a page of infinite graph paper which has its diagonals numbered consecutively with the elements of $ℤ,$ from bottom left to top right. This page has page number 0. For each $i\in \left\{±1,\dots ,±n\right\}$ place ${\text{box}}_{i}$ on diagonal ${z}_{i}$ and ${\text{box}}_{-i}$ on diagonal $-{z}_{i}\text{.}$ The boxes on each diagonal are arranged in increasing order from top left to bottom right and the placement of boxes is negative rotationally symmetric in the sense that a ${180}^{\circ }$ rotation takes boxi to ${\text{box}}_{-i}\text{.}$

Using the root system notation in (8.4),

$P(γ) = { εj-εi∣ j>iand boxiandboxj are in adjacent diagonals } and Z(γ) = { εj-εi∣ j>iand boxiandboxj are in the same diagonal } .$

If $J\subseteq P\left(\gamma \right)$ arrange the boxes on adjacent diagonals according to the rules

1. if ${\epsilon }_{j}-{\epsilon }_{i}\in J$ place ${\text{box}}_{j}$ northwest of ${\text{box}}_{i},$ and
2. if ${\epsilon }_{j}-{\epsilon }_{i}\in P\left(\gamma \right)\J$ place ${\text{box}}_{j}$ southeast of ${\text{box}}_{i}\text{.}$

A standard tableau is a negative rotationally symmetric filling $p$ of the $2n$ boxes with $-n,\dots ,-1,1,\dots ,n$ such that

1. $p\left({\text{box}}_{i}\right) if $j>i$ and ${\text{box}}_{j}$ and ${\text{box}}_{i}$ are in the same diagonal,
2. $p\left({\text{box}}_{i}\right)>p\left({\text{box}}_{j}\right)$ if $j>i,$ ${\text{box}}_{i}$ and ${\text{box}}_{j}$ are in adjacent diagonals and ${\text{box}}_{j}$ is northwest of ${\text{box}}_{i},$
3. $p\left({\text{box}}_{i}\right) if $j>i,$ ${\text{box}}_{i}$ and ${\text{box}}_{j}$ are in adjacent diagonals and ${\text{box}}_{j}$ is southeast of ${\text{box}}_{i}\text{.}$

The negative rotational symmetry means that the filling of the boxes is the same if each entry is multiplied by $-1$ and the configuration is rotated by ${180}^{\circ }\text{.}$

Example. Suppose $\gamma =\left(-2,-1,-1,-1,0,0,0,0,0,0,1,1,1,2\right)\phantom{\rule{0.2em}{0ex}}\text{and}$

$J = { ε4-ε1, ε-1-ε-4, ε4-ε2, ε-2-ε-4, ε4-ε3, ε-3-ε-2, ε5-ε1, ε-1-ε-5, ε5-ε2, ε-2-ε-5, ε5-ε3, ε-3-ε-5, ε6-ε1, ε-1-ε-6, ε6-ε2, ε-2-ε-6, ε6-ε3, ε-3-ε-6, ε7-ε6, ε-6-ε-7, ε6-ε-1, ε1-ε-6, ε5-ε-1, ε1-ε-5, ε4-ε-1, ε1-ε-4, ε5-ε-2, ε2-ε-5, ε4-ε-2, ε2-ε-4 } . = { ε4-ε1, ε4-ε2, ε4-ε3, ε5-ε1, ε5-ε2, ε5-ε3, ε6-ε1, ε6-ε2, ε6-ε3, ε7-ε6, ε6+ε1, ε5+ε1, ε5+ε2, ε4+ε1, ε4+ε2 } .$

The placed configuration of boxes corresponding to $\left(\gamma ,J\right)$ is as given below:

$0 1 1 2 0 1 0 0 -1 0 -2 -1 -1 0 -3 4 5 7 -2 6 -1 1 -6 2 -7 -5 -4 3 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 Page 0 Page 0 Page 0 contents of boxes numbering of boxes a standard tableau$

### 8.10.

A posteriori the analysis of the three cases $\beta ,$ $1/2,$ and 0, it becomes evident that the trick of using the formulation of the root system of type ${C}_{n}$ in (8.4) provides a completely uniform description of the configurations of boxes and standard tableaux corresponding to type ${C}_{n}$ local regions. All three cases give negative rotationally invariant tableaux. We could not ask for nature to work out more perfectly.

## Notes and References

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).