msPolynomialsInHLPbasis

src/Math/Algebra/Jack/Internal.hs

@@ -1039,6 +1039,55 @@ _hallLittlewoodPolynomialsInSchurBasis which lambda =
     coeffs = DM.filter (not . isZeroSpray) 
           (DM.fromDistinctDescList (zip lambdas (V.toList (getRow 1 matrix))))
 
+-- _hallLittlewoodPpolynomialInMSPbasis :: 
+--   (Eq a, AlgRing.C a) => Partition -> Map Partition (Spray a)
+-- _hallLittlewoodPpolynomialInMSPbasis lambda = 
+--   DM.unionsWith (^+^) msCombos
+--   where
+--     schurCombo = _hallLittlewoodPolynomialsInSchurBasis 'P' lambda
+--     schurAssocs = DM.assocs schurCombo
+--     msCombos = 
+--       map 
+--         (\(kappa, spray) -> 
+--           DM.mapKeys fromPartition 
+--             (DM.map (\kn -> kn .^ spray) 
+--               (kostkaNumbersWithGivenLambda (toPartitionUnsafe kappa))))
+--         schurAssocs
+
+-- | monomial symmetric polynomials in Schur polynomials basis
+msPolynomialsInSchurBasis :: 
+  Int -> Int -> Map Partition (Map Partition Rational)
+msPolynomialsInSchurBasis n weight = 
+   _inverseKostkaMatrix n weight 1 'P'
+
+msPolynomialsInHLPbasis :: 
+  Int -> Int -> Map Partition (Map Partition (Spray Rational))
+msPolynomialsInHLPbasis n weight = 
+  DM.fromDistinctAscList
+    (map (
+      \lambda -> 
+        (
+          lambda
+        , DM.filter (not . isZeroSpray) $ DM.unionsWith (^+^) (hlpCombos lambda)
+        )
+      ) lambdas)
+  where
+    msCombos = msPolynomialsInSchurBasis n weight
+    lambdas = DM.keys msCombos
+    hlpCombo mu = 
+      DM.filter (not . isZeroSpray) $ 
+        DM.fromDistinctAscList 
+          (map (\kappa -> (kappa, _kostkaFoulkesPolynomial mu kappa)) lambdas)
+    msAssocs lambda = DM.assocs (msCombos DM.! lambda)
+    hlpCombos lambda =
+      map 
+        (\(mu, r) ->
+          DM.map (\spray -> r *^ spray) (hlpCombo mu))
+        (msAssocs lambda)
+
+
+
+
 _e :: AlgRing.C a => MCP.Partition -> a -> a
 _e lambda alpha = 
   alpha * fromIntegral (_n (dualPartition lambda)) - fromIntegral (_n lambda)

src/Math/Algebra/SymmetricPolynomials.hs

@@ -910,7 +910,7 @@ jackSymbolicCombination which qspray =
 -- with symbolic parameter. 
 -- Similar to @jackSymbolicCombination@ but for a parametric spray.
 jackSymbolicCombination' :: 
-  (Eq a, AlgField.C a)
+    (Eq a, AlgField.C a)
   => Char                            -- ^ which Jack polynomials, @'J'@, @'C'@, @'P'@ or @'Q'@
   -> ParametricSpray a               -- ^ parametric spray representing a symmetric polynomial
   -> Map Partition (RatioOfSprays a) -- ^ map representing the linear combination; a partition @lambda@ in the keys of this map corresponds to the term @coeff *^ jackSymbolicPol' n lambda which@, where @coeff@ is the value attached to this key and @n@ is the number of variables of the spray
@@ -1193,7 +1193,10 @@ _tSkewSchurPolynomial f n lambda mu = sumOfSprays sprays
 
 -- | t-Schur polynomial. This is a multivariate symmetric polynomial whose 
 -- coefficients are polynomial in a single parameter usually denoted by \(t\).
--- One obtains the Schur polynomials by substituting \(t\) with \(0\). 
+-- One obtains the Schur polynomials by substituting \(t\) with \(0\).  
+-- The name \"\(t\)-Schur polynomial\" is taken from
+-- [Wheeler and Zinn-Justin's paper](https://www.sciencedirect.com/science/article/pii/S0097316518300724)
+-- /Hall polynomials, inverse Kostka polynomials and puzzles/.
 tSchurPolynomial ::
   (Eq a, AlgField.C a)
   => Int        -- ^ number of variables
@@ -1212,6 +1215,9 @@ tSchurPolynomial n lambda
 -- | t-Schur polynomial. This is a multivariate symmetric polynomial whose 
 -- coefficients are polynomial in a single parameter usually denoted by \(t\).
 -- One obtains the Schur polynomials by substituting \(t\) with \(0\). 
+-- The name \"\(t\)-Schur polynomial\" is taken from
+-- [Wheeler and Zinn-Justin's paper](https://www.sciencedirect.com/science/article/pii/S0097316518300724)
+-- /Hall polynomials, inverse Kostka polynomials and puzzles/.
 tSchurPolynomial' ::
      Int        -- ^ number of variables
   -> Partition  -- ^ integer partition