This procedure, which renames the arrow-image file (AliasBadgeIcon.icns) so that the System won't find it, affects all users and accounts and requires administrator privileges. Open Terminal and type:
<code style="color: rgb(159, 45, 0); font-family: Monaco, Courier, monospace; font-size: 12px; ">cd /System/Library/CoreServices/CoreTypes.bundle/Contents/Resources</code>Press Return and then type:
<code style="color: rgb(159, 45, 0); font-family: Monaco, Courier, monospace; font-size: 12px; ">sudo mv AliasBadgeIcon.icns AliasBadgeIcon_off.icns</code>Press Return again, type your administrator's password at the prompt, and you're done. The arrows will be gone after your next restart.
To undo this, repeat the steps above, but swap the positions of the filenames on the second command line:
<code style="color: rgb(159, 45, 0); font-family: Monaco, Courier, monospace; font-size: 12px; ">sudo mv AliasBadgeIcon_off.icns AliasBadgeIcon.icns</code>All thanks and credit to the original posters who figured this out.