How to operate the picture manipulation program ----------------------------------------------- Starting the program -------------------- To invoke the program, go to the directory containing the executable file named `pictures', or include this directory in the search path of your shell, and then type after the prompt: pictures & If you like colours, you can specify a background and foreground colour, like this: pictures -bg yellow -fg red & Building a picture ------------------ To create a picture, first select "New Diagram" from the "Functions" menu in the main window. A new window titled "Diagram 1" will appear, with no contents except a diagonal cross representing the origin. By clicking with the first (leftmost) mouse button squares can be added to the initially empty shape. The program will complete the given set of squares to the minimal skew diagram that contains them, so only the corners of the desired diagram need to be indicated. The diagram will serve as the domain of the picture to be created. Apart from adding squares, they can also be removed, by selecting "Remove BR" or "Remove TL" from the "Transformations" menu; the squares clicked at will now be removed from the shape, as well as any squares to the Bottom Right respectively to the Top Left (so as to ensure that the shape remains a skew diagram). To return to adding squares select "Add" from the "Transformations" menu. Once you are satisfied with the shape of the domain of the picture, the picture itself can be built up after selecting "Create Picture" from the "Transformations" menu. This pops up a new window called "Picture 1", which has two subwindows, one for the domain and one for the image of the picture; the domain is already indicated with the bottom left square highlighted, while the image is empty initially. The images of individual squares of the domain are defined by clicking with the first mouse button on the appropriate square in the image subwindow; at any time it is the highlighted square of the domain whose image is to be defined. Each image should satisfy the picture conditions, which is checked by the program; invalid images are refused, while giving a beep, an indication of the violated condition, and possibly highlighting (while the mouse button is depressed) the previously defined pair of squares involved in this condition. Three separate conditions should be met: the new image should be an outer cocorner of the previously defined set of images (i.e., this set sould remain a skew diagram after adding the new image, and the new image should be maximal in this set for the natural (coordinatewise) partial ordering); the image of every next square of a row should lie in a column to the right of the image of the previous square (since the squares of the domain are traversed by rows, this simply means that within each row the images should move to the right); the image of every next (higher) square of a column should lie in a row below the image of the previous square (since this image may have been defined some while ago, this condition is bit trickier than the previous one). At any time there is at least one valid square that can be chosen. The most recently entered image can be removed by selecting "Deconstruct" from the "Picture" menu, and this can be repeated in a last-in-first-out manner. When the picture is eventually completely defined, the coordinates of the originals of all the squares of the image appear, and no square of the domain is highlighted any more. Operations in the main window ----------------------------- The main window holds a record of all the diagram and picture windows that currently exist: each one has an "icon" (a rectangle containing its name) representing it within the main window. A diagram/picture window may be "iconised", in which case the icon is all that is displayed of the window; in this case a check mark appears at the right end of the icon (this is not the same as iconisation by the window manager, that might place a different kind of icon directly on the background of the screen; iconisation within the main window is the preferred way of hiding a diagram/picture window). Independently of iconisation, a diagram/picture window may be selected, which is indicated by an `S' at the left end of the icon in the main window (there can be at most one diagram and one picture selected at any one time). - Mouse actions Clicking the first mouse button in an icon iconifies the corresponding window, or if it is already iconified, deiconifies it. Clicking the second mouse button in an icon selects the corresponding window, possibly deselecting another window of the same kind. If the window was already selected, it becomes deselected. - Menus - File - Box Size This opens a "Picture settings" window (if not already open), in which the user can change the size of the squares used to draw skew diagrams. - Help This displays a window with some rudimentary information about the program. - Quit This closes all windows and terminates the program. - Functions - New Diagram This opens a fresh window for creating a skew diagram. - Create Picture This requires one of the diagram windows to be selected; it opens a new picture window in which a picture can be created with as domain the skew diagram in the selected window. - Robinson-Schensted This requires one of the picture windows to be selected; it applies the Robinson-Schensted algorithm to the picture in that window, opening a new picture window for the first picture of the result ($p$), while changing the original picture to the second picture of the result ($q$). Operations in a diagram window ------------------------------ The window can be in one of five modes: Add, Remove Bottom Right, Remove Top Left, Anti-Diagonal and Query, which can be selected from the "Transformations" menu. - Mouse actions Only clicks of the first mouse button are recognised. The effect depend on the current mode; the modes Add, Remove Bottom Right, and Remove Top Left function as described under "Building a picture" above. In the Anti-Diagonal mode several squares on one anti-diagonal are added simultaneously (this mode facilitates entry of large anti-diagonal shapes), namely the part in the positive quadrant (enclosed between the coordinate axes). The anti-diagonal containing the square clicked at is chosen (even if that square does not lie in the positive quadrant, as long as some square on its anti-diagonal does). In the Query mode the indicated square is highlighted and its coordinates displayed below, as well as a description of its position relative to the skew diagram in the window (this may help to learn the terminology used); in this mode (only) the action can be performed continuously by moving the mouse while holding down its first button. - Menus - Diagram - Select Select this diagram window (possibly deselecting another one). - Deselect Cancel selection of this diagram window. - Clear Replace the diagram in this window by an empty diagram. - Duplicate Create another diagram window identical to this one. - Iconise Remove this window from the screen and iconise it in the main window. - Done Remove this window, its contents and its icon entirely. - Transformations - Add Set Add mode. - Remove BR Set Remove Bottom Right mode - Remove TL Set Remove Top Left mode - Transpose Transpose the skew diagram. - Negate Apply central reflection in the origin to this diagram. - Anti-Diagronal Set Anti-Diagonal mode. - Query Set Query mode. - Create picture Start creating a picture with this diagram as domain. - Movement - Find Reposition window, centering on the skew diagram it contains. - Normalise Reposition window, moving origin to fixed position at top left. Operations in a picture window ------------------------------ A picture window has two subwindows, a domain window and an image window. There are two modes: creating a picture and manipulating a completed picture. Switching between these modes is automatic: the creating mode is entered when "Create Picture" is invoked from a diagram window (or from the main window), and also when "Deconstruct" or "Clear" is invoked for a completed picture; the mode swithes to completed picture mode as soon as the final image has been defined. The picture creating mode was described above under "Building a picture"; in the completed mode there is symmetry between the domain and image subwindows, and any mouse action that can be performed in one of these can also be used symmetrically in the other subwindow. - Mouse actions Clicking the first mouse button in a square of the domain or image hightlights the square and its (inverse) image, thus making the bijection visible. Like for the Query mode of diagrams, the mouse can be moved around while its button is depressed, showing the images of several squares. Clicking the first mouse button outside the domain and image of the picture only has effect if it points at a cocorner of the diagram: the operation of (domain- or image-) glissement into this square will be performed. It is possible to perform a sequence of glissements in one action by clicking the third mouse button outside the domain or image in a square that is either above and to the right (less than in the natural ordering) or below and to the left (greater than in the natural ordering) of some square of the domain or image: glissement will then be performed successively into all intermediate squares outside the domain or image (the order in which these are chosen does not affect the result). - Menus - Picture - Select Select this picture window (possibly deselecting another one). - Deselect Cancel selection of this picture window. - Clear Remove all images and start reentering a picture with same domain. - Deconstruct Remove the most recently defined image of a square. - Duplicate Create another picture window identical to this one. - Iconise Remove this window from the screen and iconise it in the main window. - Done Remove this window, its contents and its icon entirely. - Transformations - Inverse Replace picture by its inverse (interchange domain and image). - Negate Apply central reflection in the origin to domain and image of this picture. - Transpose Transpose the domain of the picture, and apply transposition combined with central symmetry in the origin to the image. - Robinson-Schensted. Apply Robinson-Schensted decomposition to this picture, creating a new picture ($p$) in which the domain is replaced by a Young diagram, while the image is replaced by another picture in which the image is replaced by the same Young diagram ($q$). - Inverse R.-S. Another picture must be selected before this operation can be performed: Apply Robinson-Schensted combination (inverse of decomposition) to this picture and the selected picture. The selected picture will remain selected and unchanged, while the current picture is replaced by the combined picture. The domain of one of the two argument picture must be an identical Young diagram to the image of the other (both combinations are allowed), otherwise the operation will not succeed (if a Young diagram appears twice as domain or twice as image one can apply "Inverse" first to get them at opposite sides). - Schutzenberger Apply the Schutzenberger involution to this picture, of which either domain or image must be a Young diagram (the shape at the other side will undergo central reflection). - Transpose both sides Apply the operation $f\mapsto f^T$ that transposes both sides of a picture. This operation can be defined in various ways, each of which involves decomposing the picture by Robinson-Schensted, then applying a combination of symmetries and/or the Schutzenberger involution to both sides of the picture, recombining the results by the inverse Robinson-Schensted algorithm, and possibly applying another symmetry. - Movement - Find domain Reposition domain subwindow, centering on the skew diagram it contains. - Find image Reposition image subwindow, centering on the skew diagram it contains. - Normalise domain Reposition domain subwindow, moving origin to fixed position at top left. - Normalise image Reposition image subwindow, moving origin to fixed position at top left. Settings window operations -------------------------- The settings (box size) window is a bit subtle to use. It reacts to keyboard input, but only of the mouse pointer is inside the inner rectangle with the number in it. This number msut first be erased by backspacing, then a new value must be entered, and when one hits (don't forget this) the change takes effect. If the size is set to less than 40 (pixels) then only the second (i.e., column) coordinate of images and inverse images are displayed in the squares, if on the other hand it is set to more than 56, then a larger font is used forr the display. Scrollbars ---------- The diagram windows and both subwindows of picture windows have rudiments of scrollbars. The only active parts are the arrows: clicking on these will move the view on the diagram, domain or image in the indicated direction. Marc van Leeuwen CWI, Amsterdam